GIFT  OF 

ASSOCIATED  ELECTRICAL   AND 
MECHANICAL   ENGINEERS 


MECHANICS  DEPARTMENT 


ELECTRIC  DISCHARGES,  WAVES  AND  IMPULSES 


Published   by  the 

Me  Grow -Hill   Book.  Company 

Ne^vYork. 

Successors  to  tKeBookDepartments  of  tKe 

McGraw  Publishing  Company  Hill  Publishing- Company 

Publishers  of  Books  for 

Elec  trical  World  The  Engineering  and  Mining'  Journal 

Engineering  Record  American   Machinist 

Electric  Railway  Journal  Coal  Age 

Metallurgical  and  Chemical  Engineering*  Power 


ELEMENTARY   LECTURES 


ON 


ELECTRIC  DISCHARGES,  WAVES 
AND  IMPULSES, 


AND 


OTHER    TRANSIENTS 


BY 

CHARLES  PROTEUS   STEINMETZ,  A.M.,  PH.D. 

\\ 

Past  President,  American  Institute  of  Electrical  Engineers 


McGRAW-HILL   BOOK  COMPANY 

239  WEST  39TH  STREET,  NEW  YOKK 

6  BOUVERIE   STREET,   LONDON,  E.G. 
1911 


r  H  . 


Library 


y 


COPYRIGHT,  1911, 

BY  THE 

McGRAW-HILL  BOOK  COMPANY 


Stanbopc  Hfress 

F.    H.  GILSON   COMPANY 
BOSTON,  U.S.A. 


PREFACE. 


IN  the  following  I  am  trying  to  give  a  short  outline  of  those 
phenomena  which  have  become  the  most  important  to  the  elec- 
trical engineer,  as  on  their  understanding  and  control  depends  the 
further  successful  advance  of  electrical  engineering.  The  art  has 
now  so  far  advanced  that  the  phenomena  of  the  steady  flow  of 
power  are  well  understood.  Generators,  motors,  transforming 
devices,  transmission  and  distribution  conductors  can,  with  rela- 
tively little  difficulty,  be. Calculated,  and  the  phenomena  occurring 
in  them  under  normal  (faa^tftmS'bf  operation  predetermined  and 
controlled.  Usually,  however,  the  limitations  of  apparatus  and 
lines  are  found  not  in  the  normal  condition  of  operation,  the  steady 
flow  of  power,  but  in  the  phenomena  occurring  under  abnormal 
though  by  no  means  unfrequent  conditions,  in  the  more  or  less 
transient  abnormal  voltages,  currents,  frequencies,  etc.;  and  the 
study  of  the  laws  of  these  transient  phenomen^4fee  electric  dis- 
charges, waves,  and  impulses,  thus  becomes  of  paramount  impor- 
tance. In  a  former  work, "  Theory  and  Calculation  of  Transient 
Electric  Phenomena  and  Oscillations,"  I  have  given  a  systematic 
study  of  these  phenomena,  as  far  as  our  present  knowledge  per- 
mits, which  by  necessity  involves  to  a  considerable  extent  the  use 
of  mathematics.  As  many  engineers  may  not  have  the  time  or 
inclination  to  a  mathematical  study,  I  have  endeavored  to  give  in 
the  following  a  descriptive  exposition  of  the  physical  nature  and 
meaning,  the  origin  and  effects,  of  these  phenomena,  with  the  use 
of  very  little  and  only  the  simplest  form  of  mathematics,  so  as  to 
afford  a  general  knowledge  of  these  phenomena  to  those  engineers 
who  have  not  the  time  to  devote  to  a  more  extensive  study, 
and  also  to  serve  as  an  introduction  to  the  study  of  "  Transient 
Phenomena."  I  have,  therefore,  in  the  following  developed  these 
phenomena  from  the  physical  conception  of  energy,  its  storage  and 
readjustment,  and  extensively  used  as  illustrations  oscillograms  of 
such  electric  discharges,  waves,  and  impulses,  taken  on  industrial 
electric  circuits  of  all  kinds,  as  to  give  the  reader  a  familiarity 


749213 


vi  PREFACE. 

with  transient  phenomena  by  the  inspection  of  their  record  on  the 
photographic  film  of  the  oscillograph.  I  would  therefore  recom- 
mend the  reading  of  the  following  pages  as  an  introduction  to 
the  study  of  "  Transient  Phenomena,"  as  the  knowledge  gained 
thereby  of  the  physical  nature  materially  assists  in  the  under- 
standing of  their  mathematical  representation,  which  latter 
obviously  is  necessary  for  their  numerical  calculation  and  pre- 
determination. 

The  book  contains  a  series  of  lectures  on  electric  discharges, 
waves,  and  impulses,  which  was  given  during  the  last  winter  to 
the  graduate  classes  of  Union  University  as  an  elementary  intro- 
duction to  and  "  translation  from  mathematics  into  English"  of  the 
"  Theory  and  Calculation  of  Transient  Electric  Phenomena  and 
Oscillations."  Hereto  has  been  added  a  chapter  on  the  calculation 
of  capacities  and  inductances  of  conductors,  since  capacity  and 
inductance  are  the  fundamental  quantities  on  which  the  transients 
depend. 

In  the  preparation  of  the  work,  I  have  been  materially  assisted 
by  Mr.  C.  M.  Davis,  M.E.E.,  who  kindly  corrected  and  edited 
the  manuscript  and  illustrations,  and  to  whom  I  wish  to  express 
my  thanks. 

CHARLES   PROTEUS  STEINMETZ. 

October,  1911. 


CONTENTS. 


PAGE 

LECTURE  I.  —  NATURE  AND  ORIGIN  OF  TRANSIENTS 1 

1 .  Electric  power  and  energy.    Permanent  and  transient  phenomena. 
Instance  of  permanent  phenomenon;  of  transient;  of  combination 
of  both.     Transient  as  intermediary  condition  between  permanents. 

2.  Energy  storage  in  electric  circuit,  by  magnetic  and  dielectric  field. 
Other  energy  storage.     Change  of  stored  energy  as  origin  of  tran- 
sient. 

3.  Transients  existing  with  all  forms  of  energy:  transients  of  rail- 
way car;  of  fan  motor;  of  incandescent  lamp.     Destructive  values. 
High-speed   water-power  governing.      Fundamental   condition  of 
transient.     Electric   transients  simpler,   their  theory  further  ad- 
vanced, of  more  direct  industrial  importance. 

4.  Simplest  transients:  proportionality  of  cause  and  effect.     Most 
electrical  transients  of  this  character.     Discussion  of  simple  tran- 
sient of  electric  circuit.     Exponential  function  as  its  expression. 
Coefficient  of  its  exponent.     Other  transients:  deceleration  of  ship. 

5.  Two    classes    of    transients:    single-energy    and    double-energy 
transients.      Instance  of  car  acceleration;  of  low- voltage  circuit; 
of  pendulum;  of  condenser  discharge  through  inductive    circuit. 
Transients  of  more  than  two  forms  of  energy. 

6.  Permanent  phenomena  usually  simpler  than  transients.     Re- 
duction of  alternating-current  phenomena  to  permanents  by  effec- 
tive values  and  by  symbolic  method.     Nonperiodic  transients. 

LECTURE  II.  —  THE  ELECTRIC  FIELD 10 

7.  Phenomena  of  electric  power  flow:  power  dissipation  in  con- 
ductor; electric  field  consisting  of  magnetic  field  surrounding  con- 
ductor and  electrostatic  or  dielectric  field  issuing  from  conductor. 
Lines  of  magnetic  force;  lines  of  dielectric  force. 

8.  The   magnetic   flux,   inductance,   inductance  voltage,  and  the 
energy  of  the  magnetic  field. 

9.  The  dielectric  flux,  capacity,  capacity  current,  and  the  energy 
of  the  dielectric  field.     The  conception  of  quantity  of  electricity, 
electrostatic  charge  and  condenser;  the  conception  of  quantity  of 
magnetism. 

10.  Magnetic  circuit  and  dielectric  circuit.     Magnetomotive  force, 
magnetizing  force,  magnetic  field  intensity,  and  magnetic  density. 
Permeability.      Magnetic  materials. 

vii 


Vlll  CONTENTS. 

PAGE 

11.  Electromotive  force,    electrifying   force   or   voltage   gradient. 
Dielectric  field  intensity  and  dielectric  density.     Specific  capacity 
or  permittivity.     Velocity  of  propagation. 

12.  Tabulation  of  corresponding  terms  of  magnetic  and  of  die- 
lectric field.     Tabulation  of  analogous  terms  of  magnetic,  dielec- 
tric, and  electric  circuit. 

LECTURE  III.  —  SINGLE-ENERGY   TRANSIENTS   IN  CONTINUOUS-CUR- 
RENT CIRCUITS 19 

13.  Single-energy    transient   represents    increase    or   decrease   of 
energy.     Magnetic  transients  of  low-  and  medium-voltage  circuits. 
Single-energy  and   double-energy  transients  of   capacity.     Discus- 
sion of  the  transients  of  4>,  i,  e,  of  inductive  circuit.     Exponen- 
tial equation.     Duration  of  the  transient,  time  constant.     Numer- 
ical values  of  transient  of  intensity  1  and  duration  1.     The  three 
forms  of  the  equation  of  the  magnetic   transient.     Simplification 
by  choosing  the  starting  moment  as  zero  of  time. 

14.  Instance  of  the  magnetic  transient  of  a  motor  field.     Calcula- 
tion of  its  duration. 

15.  Effect  of  the  insertion  of  resistance  on  voltage  and  duration  of 
the  magnetic  transient.     The  opening  of  inductive  circuit.     The 
effect  of  the  opening  arc  at  the  switch. 

16.  The  magnetic  transient  of  closing  an  inductive  circuit.     General 
method  of  separation  of  transient  and  of  permanent  terms  during 
the  transition  period. 

LECTURE  IV.  —  SINGLE-ENERGY    TRANSIENTS    OF    ALTERNATING-CUR- 
RENT CIRCUITS 30 

17.  Separation  of  current  into  permanent  and  transient  component. 
Condition  of  maximum  and  of  zero  transient.     The  starting  of  an 
alternating  current;  dependence  of  the  transient  on  the  phase;  maxi- 
mum and  zero  value. 

18.  The  starting  transient  of  the  balanced  three-phase  system. 
Relation  between  the  transients  of  the  three  phases.     Starting 
transient  of  three-phase  magnetic  field,  and  its  construction.     The 
oscillatory  start  of  the  rotating  field.     Its  independence  of  the  phase 
at  the  moment  of  start.     Maximum  value  of  rotating-field  tran- 
sient, and  its  industrial  bearing. 

19.  Momentary  short-circuit  current  of    synchronous  alternator, 
and  current  rush  in  its  field  circuit.     Relation  between  voltage, 
load,  magnetic  field  flux,  armature  reaction,  self-inductive  reactance, 
and  synchronous  reactance  of  alternator.     Ratio  of  momentary  to 
permanent  short-cicurit  current. 

20.  The  magnetic  field  transient  at  short  circuit  of  alternator.     Its 
effect  on  the  armature  currents,  and  on  the  field  current.     Numeri- 
cal relation  bet  ween  the  transients  of  magnetic  flux,  armature  currents, 
armature  reaction,  and  field  current.     The  starting  transient  of  the 
armature  currents.    The  transient  full-frequency  pulsation  of  the 


CONTENTS.  ix 

PAGE 

field  current  caused  by  it.  Effect  of  inductance  in  the  exciter  field. 
Calculation  and  construction  of  the  transient  phenomena  of  a  poly- 
phase alternator  short  circuit. 

21.  The  transients  of  the  single-phase  alternator  short  circuit. 
The  permanent  double- frequency  pulsation  of  armature  reaction 
and  of  field  current.     The  armature  transient  depending  on  the 
phase  of  the  wave.     Combination  of  full-frequency  transient  and 
double-frequency  permanent  pulsation  of    field  current,  and  the 
shape  of  the  field  current  resulting  therefrom.     Potential  difference 
at  field  terminal  at  short  circuit,  and  its  industrial  bearing. 

LECTURE  V.  —  SINGLE-ENERGY  TRANSIENT  OF  IRONCLAD  CIRCUIT.  ...       52 

22.  Absence   of   proportionality   between    current   and   magnetic 
flux  in  ironclad  circuit.       Numerical  calculation  by  step-by-step 
method.     Approximation  of  magnetic  characteristic  by  Frohlich's 
formula,  and  its  rationality. 

23.  General  expression  of  magnetic  flux  in  ironclad  circuit.     Its 
introduction  in  the  differential  equation  of  the  transient.     Integra- 
tion, and  calculation  of  a  numerical  instance.     High-current  values 
and  steepness  of  ironclad  magnetic  transient,  and  its  industrial 
bearing. 

LECTURE  VI.  —  DOUBLE-ENERGY  TRANSIENTS 59 

24.  Single-energy  transient,  after  separation  from  permanent  term, 
as  a  steady  decrease  of  energy.     Double-energy  transient  consisting 
of  energy-dissipation  factor  and  energy-transfer  factor.     The  latter 
periodic  or  unidirectional.  The  latter  rarely  of  industrial  importance. 

25.  Pulsation  of  energy  during  transient.     Relation  between  maxi- 
mum current  and  maximum  voltage.     The  natural  impedance  and 
the  natural  admittance  of  the  circuit.     Calculation  of  maximum 
voltage  from  maximum  current,  and  inversely.     Instances  of  line 
short  circuit,  ground  on  cable,  lightning  stroke.     Relative  values  of 
transient  currents  and  voltages  in  different  classes  of  circuits. 

26.  Trigonometric  functions  of  the  periodic  factor  of  the  transient. 
Calculation  of  the  frequency.    Initial  values  of  current  and  voltage. 

27.  The  power-dissipation  factor  of  the  transient.     Duration  of  the 
double-energy  transient  the  harmonic  mean  of  the  duration  of  the 
magnetic  and  of  the  dielectric  transient.     The  dissipation  expo- 
nent, and  its  usual  approximation.     The  complete  equation  of  the 
double-energy  transient.     Calculation  of  numerical  instance. 

LECTURE  VII.  —  LINE  OSCILLATIONS 72 

28.  Review  of  the  characteristics  of  the  double-energy  transient: 
periodic  and  transient  factor;  relation  between  current  and  voltage; 
the  periodic  component  and  the  frequency;  the  transient  compo- 
nent and  the  duration;  the  initial  values  of  current  and  voltage. 


X  CONTENTS. 

PAGE 

Modification  for  distributed  capacity  and  inductance:  the  distance 
phase  angle  and  the  velocity  of  propagation;  the  time  phase  angle; 
the  two  forms  of  the  equation  of  the  line  oscillation. 

29.  Effective  inductance  and  effective  capacity,  and  the  frequency 
of  the  line  oscillation.     The  wave  length.     The  oscillating-line  sec- 
tion as  quarter  wave  length. 

30.  Relation  between  inductance,  capacity,  and  frequency  of  prop- 
agation.    Importance  of  this  relation  for  calculation  of  line  con- 
stants. 

31.  The  different  frequencies  and  wave  lengths  of  the  quarter- 
wave  oscillation;  of  the  half- wave  oscillation. 

32.  The  velocity  unit  of  length.     Its  importance  in  compound 
circuits.     Period,   frequency,   time,  and  distance  angles,  and  the 
general  expression  of  the  line  oscillation. 

LECTURE  VIII.  —  TRAVELING  WAVES 88 

33.  The  power  of  the  stationary  oscillation  and  its  correspondence 
with  reactive  power  of  alternating  currents.     The  traveling  wave 
and  its  correspondence  with  effective  power  of  alternating  currents. 
Occurrence  of  traveling  waves:  the  lightning  stroke:     The  traveling 
wave  of  the  compound  circuit. 

34.  The  flow  of  transient  power  and  its  equation.     The  power- 
dissipation  constant  and  the  power-transfer  constant.      Increasing 
and  decreasing  power  flow  in  the  traveling  wave.     The  general 
equation  of  the  traveling  wave. 

35.  Positive  and  negative  power- transfer  constants.      Undamped 
oscillation  and  cumulative  oscillation.     The  arc  as  their  source. 
The  alternating-current  transmission-line  equation  as  special  case  of 
traveling  wave  of  negative  power-transfer  constant. 

36.  Coexistence  and  combination  of  traveling  waves  and  stationary 
oscillations.     Difference   from    effective   and   reactive   alternating 
waves.     Industrial    importance    of    traveling    waves.     Their    fre- 
quencies.    Estimation  of  their  effective  frequency  if  very  high. 

37.  The  impulse  as  traveling  wave.     Its  equations.     The  wave 
front. 

LECTURE  IX.  —  OSCILLATIONS  OF  THE  COMPOUND  CIRCUIT 108 

38.  The    stationary    oscillation    of    the    compound    circuit.     The 
time  decrement  of  the  total  circuit,  and  the  power-dissipation  and 
power-transfer  constants  of  its  section.     Power  supply  from  section 
of  low-energy  dissipation  to  section  of  high-energy  dissipation. 

39.  Instance  of  oscillation  of  a  closed   compound   circuit.     The 
two  traveling  waves  and  the  resultant  transient-power  diagram. 

40.  Comparison  of  the  transient-power  diagram  with  the  power 
diagram  of  an  alternating- current  circuit.      The  cause  of  power 
increase  in  the  line.     The  stationary  oscillation  of  an  open  com- 
pound circuit. 


CONTENTS.  xi 

PAGE 

41.  Voltage  and  current  relation  between  the  sections  of  a  compound 
oscillating  circuit.     The  voltage  and  current  transformation  at  the 
transition  points  between  circuit  sections. 

42.  Change  of  phase  angle  at  the  transition  points  between  sec- 
tions of  a  compound  oscillating  circuit.     Partial  reflection  at  the 
transition  point. 

LECTURE  X.  —  INDUCTANCE  AND  CAPACITY  OF  ROUND  PARALLEL  CON- 
DUCTORS      119 

43.  Definition  of  inductance  and  of  capacity.     The  magnetic  and 
the  dielectric  field.     The  law  of  superposition  of  fields,  and  its  use 
for  calculation. 

44.  Calculation  of  inductance  of  two  parallel  round  conductors. 
External  magnetic  flux  and  internal  magnetic  flux. 

45.  Calculation  and  discussion  of  the  inductance  of  two  parallel 
conductors  at  small  distances  from  each  other.     Approximations 
and  their  practical  limitations. 

46.  Calculation  of  capacity  of  parallel  conductors  by  superposition 
of  dielectric  fields.     Reduction  to  electromagnetic  units  by  the 
velocity    of   light.     Relation   between  inductance,    capacity,   and 
velocity  of  propagation. 

47.  Conductor    with    ground    return,    inductance,   and    capacity. 
The  image  conductor.     Limitations  of  its  application.     Correction 
for  penetration  of  return  current  in  ground. 

48.  Mutual  inductance  between  circuits.     Calculation  of  equation, 
and  approximation. 

49.  Mutual  capacity  between  circuits.      Symmetrical  circuits  and 
asymmetrical  circuits.      Grounded  circuit. 

50.  The  three-phase  circuit.     Inductance  and   capacity  of  two- 
wire  single-phase  circuit,  of  single-wire  circuit  with  ground  return, 
and  of  three-wire  three-phase  circuit.     Asymmetrical  arrangement 
of  three-phase  circuit.     Mutual  inductance  and  mutual  capacity 
with  three-phase  circuit. 


ELEMENTAEY    LECTURES    ON    ELECTEIC 

DISCHARGES,  WAVES  AND  IMPULSES, 

AND   OTHER  TRANSIENTS. 

LECTURE  I. 

NATURE   AND    ORIGIN   OF  TRANSIENTS. 

i.  Electrical  engineering  deals  with  electric  energy  and  its 
flow,  that  is,  electric  power.  Two  classes  of  phenomena  are  met: 
permanent  and  transient,  phenomena.  To  illustrate:  Let  G  in 
Fig.  1  be  a  direct-current  generator,  which  over  a  circuit  A  con- 
nects to  a  load  L,  as  a  number  of  lamps,  etc.  In  the  generator 
G,  the  line  A,  and  the  load  L,  a  current  i  flows,  and  voltages  e 


Fig.  1. 

exist,  which  are  constant,  or  permanent,  as  long  as  the  conditions 
of  the  circuit  remain  the  same.  If  we  connect  in  some  more 
lights,  or  disconnect  some  of  the  load,  we  get  a  different  current 
i',  and  possibly  different  voltages  e1 ';  but  again  i'  and  e'  are  per- 
manent, that  is,  remain  the  same  as  long  as  the  circuit  remains 
unchanged. 

Let,  however,  in  Fig.  2,  a  direct-current  generator  G  be  connected 
to  an  electrostatic  condenser  C.  Before  the  switch  S  is  closed,  and 
therefore  also  in  the  moment  of  closing  the  switch,  no  current  flows 
in  the  line  A.  Immediately  after  the  switch  S  is  closed,  current 
begins  to  flow  over  line  A  into  the  condenser  C,  charging  this 
condenser  up  to  the  voltage  given  by  the  generator.  When  the 

1 


DISCHARGES,  WAVES  AND  IMPULSES. 

condenser  C  is  charged,  the  current  in  the  line  A  and  the  condenser 
C  is  zero  again.  That  is,  the  permanent  condition  before  closing 
the  switch  S,  and  also  some  time  after  the  closing  of  the  switch, 
is  zero  current  in  the  line.  Immediately  after  the  closing  of 
the  switch,  however,  current  flows  for  a  more  or  less  short  time. 
With  the  condition  of  the  circuit  unchanged:  the  same  generator 
voltage,  the  switch  S  closed  on  the  same  circuit,  the  current 
nevertheless  changes,  increasing  from  zero,  at  the  moment  of 
closing  the  switch  S,  to  a  maximum,  and  then  decreasing  again  to 
zero,  while  the  condenser  charges  from  zero  voltage  to  the  genera- 
tor voltage.  We  then  here  meet  a  transient  phenomenon,  in  the 
charge  of  the  condenser  from  a  source  of  continuous  voltage. 


Commonly,  transient  and  permanent  phenomena  are  super- 
imposed upon  each  other.  For  instance,  if  in  the  circuit  Fig.  1 
we  close  the  switch  S  connecting  a  fan  motor  F,  at  the  moment  of 
closing  the  switch  S  the  current  in  the  fan-motor  circuit  is  zero. 
It  rapidly  rises  to  a  maximum,  the  motor  starts,  its  speed  increases 
while  the  current  decreases,  until  finally  speed  and  current  become 
constant;  that  is,  the  permanent  condition  is  reached. 

The  transient,  therefore,  appears  as  intermediate  between  two 
permanent  conditions:  in  the  above  instance,  the  fan  motor  dis- 
connected, and  the  fan  motor  running  at  full  speed.  The  question 
then  arises,  why  the  effect  of  a  change  in  the  conditions  of  an 
electric  circuit  does  not  appear  instantaneously,  but  only  after  a 
transition  period,  requiring  a  finite,  though  frequently  very  short, 
time. 

2.  Consider  the  simplest  case:  an  electric  power  transmission 
(Fig.  3).  In  the  generator  G  electric  power  is  produced  from  me- 
chanical power,  and  supplied  to  the  line  A .  In  the  line  A  some  of 
this  power  is  dissipated,  the  rest  transmitted  into  the  load  L, 
where  the  power  is  used.  The  consideration  of  the  electric  power 


NATURE  AND  ORIGIN  OF  TRANSIENTS.  3 

in  generator,  line,  and  load  does  not  represent  the  entire  phenome- 
non. While  electric  power  flows  over  the  line  A ,  there  is  a  magnetic 
field  surrounding  the  line  conductors,  and  an  electrostatic  field 
issuing  from  the  line  conductors.  The  magnetic  field  and  the 
electrostatic  or  "dielectric  "  field  represent  stored  energy.  Thus, 
during  the  permanent  conditions  of  the  flow  of  power  through  the 
circuit  Fig.  3,  there  is  electric  energy  stored  in  the  space  surround- 
ing the  line  conductors.  There  is  energy  stored  also  in  the  genera- 
tor and  in  the  load ;  for  instance,  the  mechanical  momentum  of  the 
revolving  fan  in  Fig.  1,  and  the  heat  energy  of  the  incandescent 
lamp  filaments.  The  permanent  condition  of  the  circuit  Fig.  3 
thus  represents  not  only  flow  of  power,  but  also  storage  of  energy. 
When  the  switch  S  is  open,  and  no  power  flows,  no  energy  is 
stored  in  the  system.  If  we  now  close  the  switch,  before  the 
permanent  condition  corresponding  to  the  closed  switch  can  occur, 


Fig.  3. 

the  stored  energy  has  to  be  supplied  from  the  source  of  power;  that 
is,  for  a  short  time  power,  in  supplying  the  stored  energy,  flows  not 
only  through  the  circuit,  but  also  from  the  circuit  into  the  space 
surrounding  the  conductors,  etc.  This  flow  of  power,  which  sup- 
plies the  energy  stored  in  the  permanent  condition  of  the  circuit, 
must  cease  as  soon  as  the  stored  energy  has  been  supplied,  and 
thus  is  a  transient. 

Inversely,  if  we  disconnect  some  of  the  load  L  in  Fig.  3,  and 
thereby  reduce  the  flow  of  power,  a  smaller  amount  of  stored 
energy  would  correspond  to  that  lesser  flow,  and  before  the 
conditions  of  the  circuit  can  become  stationary,  or  permanent 
(corresponding  to  the  lessened  flow  of  power),  some  of  the  stored 
energy  has  to  be  returned  to  the  circuit,  or  dissipated,  by  a 
transient. 

Thus  the  transient  is  the  result  of  the  change  of  the  amount  of 
stored  energy,  required  by  the  change  of  circuit  conditions,  and 


4          ELECTRIC  DISCHARGES,  WAVES  AND  IMPULSES. 

is  the  phenomenon  by  which  the  circuit  readjusts  itself  to  the 
change  of  stored  energy.  It  may  thus  be  said  that  the  perma- 
nent phenomena  are  the  phenomena  of  electric  power,  the  tran- 
sients the  phenomena  of  electric  energy. 

3.  It  is  obvious,  then,  that  transients  are  not  specifically  electri- 
cal phenomena,  but  occur  with  all  forms  of  energy,  under  all  condi- 
tions where  energy  storage  takes  place. 

Thus,  when  we  start  the  motors  propelling  an  electric  car,  a 
transient  period,  of  acceleration,  appears  between  the  previous 
permanent  condition  of  standstill  and  the  final  permanent  con- 
dition of  constant-speed  running;  when  we  shut  off  the  motors, 
the  permanent  condition  of  standstill  is  not  reached  instantly, 
but  a  transient  condition  of  deceleration  intervenes.  When  we 
open  the  water  gates  leading  to  an  empty  canal,  a  transient 
condition~"of  flow  and  water  level  intervenes  while  the  canal  is 
filling,  until  the  permanent  condition  is  reached.  Thus  in  the  case 
of  the  fan  motor  in  instance  Fig.  1,  a  transient  period  of  speed 
and  mechanical  energy  appeared  while  the  motor  was  speeding  up 
and  gathering  the  mechanical  energy  of  its  momentum.  When 
turning  on  an  incandescent  lamp,  the  filament  passes  a  transient 
of  gradually  rising  temperature. 

Just  as  electrical  transients  may,  under  certain  conditions,  rise 
to  destructive  values;  so  transients  of  other  forms  of  energy  may 
become  destructive,  or  may  require  serious  consideration,  as,  for 
instance,  is  the  case  in  governing  high-head  water  powers.  The 
column  of  water  in  the  supply  pipe  represents  a  considerable 
amount  of  stored  mechanical  energy,  when  flowing  at  velocity, 
under  load.  If,  then,  full  load  is  suddenly  thrown  off,  it  is  not 
possible  to  suddenly  stop  the  flow  of  water,  since  a  rapid  stopping 
would  lead  to  a  pressure  transient  of  destructive  value,  that  is, 
burst  the  pipe.  Hence  the  use  of  surge  tanks,  relief  valves,  or 
deflecting  nozzle  governors.  Inversely,  if  a  heavy  load  comes  on 
suddenly,  opening  the  nozzle  wide  does  not  immediately  take  care 
of  the  load,  but  momentarily  drops  the  water  pressure  at  the 
nozzle,  while  gradually  the  water  column  acquires  velocity,  that 
is,  stores  energy. 

The  fundamental  condition  of  the  appearance  of  a  transient 
thus  is  such  a  disposition  of  the  stored  energy  in  the  system  as 
differs  from  that  required  by  the  existing  conditions  of  the  system; 
and  any  change  of  the  condition  of  a  system,  which  requires  a 


NATURE  AND  ORIGIN  OF  TRANSIENTS.  O 

change  of  the  stored  energy,  of  whatever  form  this  energy  may  be, 
leads  to  a  transient. 

Electrical  transients  have  been  studied  more  than  transients  of 
other  forms  of  energy  because : 

(a)  Electrical  transients  generally  are  simpler  in  nature,  and 
therefore  yield  more  easily  to   a  theoretical   and  experimental 
investigation. 

(b)  The  theoretical   side   of   electrical   engineering  is   further 
advanced  than  the  theoretical  side  of  most  other  sciences,  and 
especially : 

(c)  The  destructive  or  harmful  effects  of  transients  in  electrical 
systems  are  far  more  common  and  more  serious  than  with  other 
forms  of  energy,  and  the  engineers  have  therefore  been  driven  by 
necessity  to  their  careful  and  extensive  study. 

4.  The  simplest  form  of  transient  occurs  where  the  effect  is 
directly  proportional  to  the  cause.  This  is  generally  the  case  in 
electric  circuits,  since  voltage,  current,  magnetic  flux,  etc.,  are 
proportional  to  each  other,  and  the  electrical  transients  therefore 
are  usually  of  the  simplest  nature.  In  those  cases,  however, 
where  this  direct  proportionality  does  not  exist,  as  for  instance  in 
inductive  circuits  containing  iron,  or  in  electrostatic  fields  exceed- 
ing the  corona  voltage,  the  transients  also  are  far  more  complex, 
and  very  little  work  has  been  done,  and  very  little  is  known,  on 
these  more  complex  electrical  transients. 

Assume  that  in  an  electric  circuit  we  have  a  transient  cur- 
rent, as  represented  by  curve  i  in  Fig.  4 ;  that  is,  some  change  of 
circuit  condition  requires  a  readjustment  of  the  stored  energy, 
which  occurs  by  the  flow  of  transient  current  i.  This  current 
starts  at  the  value  ii,  and  gradually  dies  down  to  zero.  Assume 
now  that  the  law  of  proportionality  between  cause  and  effect 
applies;  that  is,  if  the  transient  current  started  with  a  different 
value,  izj  it  would  traverse  a  curve  if,  which  is  the  same  as  curve 
i,  except  that  all  values  are  changed  proportionally,  by  the  ratio 

^;  that  is,  i'=iX*- 
iij  ii 

Starting  with  current  ii,  the  transient  follows  the  curve  i; 
starting  with  z'2,  the  transient  follows  the  proportional  curve  i' . 
At  some  time,  t,  however,  the  current  i  has  dropped  to  the  value  t'2, 
with  which  the  curve  i'  started.  At  this  moment  t,  the  conditions 
in  the  first  case,  of  current  i,  are  the  same  as  the  conditions  in 


6 


ELECTRIC  DISCHARGES,  WAVES  AND  IMPULSES. 


the  second  case,  of  current  if,  at  the  moment  t\;  that  is,  from  t 
onward,  curve  i  is  the  same  as  curve  i'  from  time  i\  onward.     Since 


t! 

Fig.  4.  —  Curve  of  Simple  Transient:  Decay  of  Current. 


if  is  proportional  to  i  from  any  point  t  onward,  curve  i'  is  propor- 
tional to  the  same  curve  i  from  t\  onward.     Hence,  at  time  t\,  it  is 

diz      dii  ^.  i% 
dti      dti      ii 

But  since  -^  and  i2  at  t\  are  the  same  as  -r  and  i  at  time  t,  it 

CLL\  dv 

follows: 

di      dii  i 


or, 


di 


where  c  =  —  -  -r;1  =  constant,  and  the  minus  sign  is  chosen,  as 
ii  at 

di  . 

-r  is  negative. 

at 

As  in  Fig.  4: 


~aJi  =  ii, 
1^  dii  _  tan  <f>  _  _1_ . 

~  ~  ~       ' 


NATURE  AND  ORIGIN  OF  TRANSIENTS.  7 

that  is,  c  is  the  reciprocal  of  the  projection  T  =  tj*  on  the  zero  line 
of  the  tangent  at  the  starting  moment  of  the  transient. 

Since 

di  , 

—  =  —  cdt; 

that  is,  the  percentual  change  of  current  is  constant,  or  in  other 
words,  in  the  same  time,  the  current  always  decreases  by  the  same 
fraction  of  its  value,  no  matter  what  this  value  is. 
Integrated,  this  equation  gives: 

log  i  =  —  ct  +  C, 
i  =  Ae~ct, 

or>  i~#€:':*5 

that  is,  the  curve  is  the  exponential. 

The  exponential  curve  thus  is  the  expression  of  the  simplest 
form  of  transient.  This  explains  its  common  occurrence  in  elec- 
trical and' other  transients.  Consider,  for  instance,  the  decay  of 
radioactive  substances :  the  radiation,  which  represents  the  decay, 

is  proportional  to  the  amount  of  radiating  material;  it  is  ~-r-  =  cm, 

Cit 

which  leads  to  the  same  exponential  function. 

Not  all  transients,  however,  are  of  this  simplest  form.  For 
instance,  the  deceleration  of  a  ship  does  not  follow  the  exponential, 
but  at  high  velocities  the  decrease  of  speed  is  a  greater  fraction  of 
the  speed  than  during  the  same  time  interval  at  lower  velocities, 
and  the  speed-time  curves  for  different  initial  speeds  are  not  pro- 
portional to  each  other,  but  are  as  shown  in  Fig.  5.  The  reason 
is,  that  the  frictional  resistance  is  not  proportional  to  the  speed, 
but  to  the  square  of  the  speed. 

5.   Two  classes  of  transients  may  occur: 

1.  Energy  may  be  stored  in  one  form  only,  and  the  only  energy 
change  which  can  occur  thus  is  an  increase  or  a  decrease  of  the 
stored  energy. 

2.  Energy  may  be  stored  in  two  or  more  different  forms,  and  the 
possible  energy  changes  thus  are  an  increase  or  decrease  of  the 
total  stored  energy,  or  a  change  of  the  stored  energy  from  one  form 
to  another.     Usually  both  occur  simultaneously. 

An  instance  of  the  first  case  is  the  acceleration  or  deceleration 


8 


ELECTRIC  DISCHARGES,   WAVES  AND  IMPULSES. 


of  a  train,  or  a  ship,  etc. :  here  energy  can  be  stored  only  as  mechan- 
ical momentum,  and  the  transient  thus  consists  of  an  increase  of 
the  stored  energy,  during  acceleration,  or  of  a  decrease,"  during 


0 10    20     30     40 


Seconds 
50      60     70     80     90    100    110    120 


Fig.  5.  —  Deceleration  of  Ship. 

deceleration.  Thus  also  in  a  low-voltage  electric  circuit  of  negli- 
gible capacity,  energy  can  be  stored  only  in  the  magnetic  field,  and 
the  transient  represents  an  increase  of  the  stored  magnetic  energy, 
during  increase  of  current,  or  a  decrease  of  the  magnetic  energy, 
during  a  decrease  of  current. 

An  instance  of  the  second  case  is  the  pendulum,  Fig.  6 :  with  the 
weight  at  rest  in  maximum  elevation,  all  the  stored  energy  is 

potential  energy  of  gravita- 
tion. This  energy  changes  to 
kinetic  mechanical  energy  until 
in  the  lowest  position,  a,  when 
all  the  potential  gravitational 
energy  has  been  either  con- 
verted to  kinetic  mechanical 
energy  or  dissipated.  Then, 
during  the  rise  of  the  weight, 
that  part  of  the  energy  which 
is  not  dissipated  again  changes 
to  potential  gravitational  en- 
ergy, at  c,  then  back  again  to 
kinetic  energy,  at  a;  and  in  this  manner  the  total  stored  energy 
is  gradually  dissipated,  by  a  series  of  successive  oscillations  or 
changes  between  potential  gravitational  and  kinetic  mechanical 


Fig.  6. 


Double-energy  Transient 
of  Pendulum. 


NATURE  AND  ORIGIN  OF  TRANSIENTS. 

energy.  Thus  in  electric  circuits  containing  energy  stored  in  the 
magnetic  and  in  the  dielectric  field,  the  change  of  the  amount 
of  stored  energy  —  decrease  or  increase  —  frequently  occurs  by  a 
series  of  successive  changes  from  magnetic  to  dielectric  and  back 
again  from  dielectric  to  magnetic  stored  energy.  This  for  instance 
is  the  case  in  the  charge  or  discharge  of  a  condenser  through  an 
inductive  circuit. 

If  energy  can  be  stored  in  more  than  two  different  forms,  still 
more  complex  phenomena  may  occur,  as  for  instance  in  the  hunt- 
ing of  synchronous  machines  at  the  end  of  long  transmission  lines, 
where  energy  can  be  stored  as  magnetic  energy  in  the  line  and 
apparatus,  as  dielectric  energy  in  the  line,  and  as  mechanical 
energy  in  the  momentum  of  the  motor. 

6.  The  study  and  calculation  of  the  permanent  phenomena  in 
electric  circuits  are  usually  far  simpler  than  are  the  study  and 
calculation  of  transient  phenomena.  However,  only  the  phe- 
nomena of  a  continuous-current  circuit  are  really  permanent. 
The  alternating-current  phenomena  are  transient,  as  the  e.m.f. 
continuously  and  periodically  changes,  and  with  it  the  current, 
the  stored  energy,  etc.  The  theory  of  alternating-current  phe- 
nomena, as  periodic  transients,  thus  has  been  more  difficult  than 
that  of  continuous-current  phenomena,  until  methods  were  devised 
to  treat  the  periodic  transients  of  the  alternating-current  circuit 
as  permanent  phenomena,  by  the  conception  of  the  "  effective 
values,"  and  more  completely  by  the  introduction  of  the  general 
number  or  complex  quantity,  which  represents  the  periodic  func- 
tion of  time  by  a  constant  numerical  value.  In  this  feature  lies 
the  advantage  and  the  power  of  the  symbolic  method  of  dealing 
with  alternating-current  phenomena,  —  the  reduction  of  a  periodic 
transient  to  a  permanent  or  constant  quantity.  For  this  reason, 
wherever  periodic  transients  occur,  as  in  rectification,  commuta- 
tion, etc.,  a  considerable  advantage  is  frequently  gained  by  their 
reduction  to  permanent  phenomena,  by  the  introduction  of  the 
symbolic  expression  of  the  equivalent  sine  wave. 

Hereby  most  of  the  periodic  transients  have  been  eliminated 
from  consideration,  and  there  remain  mainly  the  nonperiodic 
transients,  as  occur  at  any  change  of  circuit  conditions.  Since 
they  are  the  phenomena  of  the  readjustment  of  stored  energy,  a 
study  of  the  energy  storage  of  the  electric  circuit,  that  is,  of  its 
magnetic  and  dielectric  field,  is  of  first  importance. 


LECTURE  II. 

THE  ELECTRIC   FIELD. 

7.   Let,  in  Fig.  7,  a  generator  G  transmit  electric  power  over 
line  A  into  a  receiving  circuit  L. 

While  power  flows  through 
the  conductors  A,  power  is  con- 
sumed in  these  conductors  by 
conversion  into  heat,  repre- 

sented  by  i?r.     This,  however, 

Fig.  7.  is   not   all,   but   in  the    space 

surrounding  the  conductor  cer- 
tain phenomena  occur:  magnetic  and  electrostatic  forces  appear. 


Fig.  8.  —  Electric  Field  of  Conductor. 

The  conductor  is  surrounded  by  a  magnetic  field,  or  a  magnetic 
flux,  which  is  measured  by  the  number  of  lines  of  magnetic  force  <J>. 
With  a  single  conductor,  the  lines  of  magnetic  force  are  concentric 
circles,  as  shown  in  Fig.  8.  By  the  return  conductor,  the  circles 

10 


THE  ELECTRIC  FIELD. 


11 


are  crowded  together  between  the  conductors,  and  the  magnetic 
field  consists  of  eccentric  circles  surrounding  the  conductors,  as 
shown  by  the  drawn  lines  in  Fig.  9. 

An  electrostatic,  or,  as  more  properly  called,  dielectric  field,  issues 
from  the  conductors,  that  is,  a  dielectric  flux  passes  between  the 
conductors,  which  is  measured  by  the  number  of  lines  of  dielectric 
force  ty.  With  a  single  conductor,  the  lines  of  dielectric  force  are 
radial  straight  lines,  as  shown  dotted  in  Fig.  8.  By  the  return 
conductor,  they  are  crowded  together  between  the  conductors, 
and  form  arcs  of  circles,  passing  from  conductor  to  return  conduc- 
tor, as  shown  dotted  in  Fig.  9. 


Fig.  9.  —  Electric  Field  of  Circuit. 

The  magnetic  and  the  dielectric  field  of  the  conductors  both  are 
included  in  the  term  electric  field,  and  are  the  two  components  of 
the  electric  field  of  the  conductor. 

8.  The  magnetic  field  or  magnetic  flux  of  the  circuit,  <£,  is  pro- 
portional to  the  current,  i,  with  a  proportionality  factor,  L,  which 
is  called  the  inductance  of  the  circuit. 


=  Li. 


(1) 


The  magnetic  field  represents  stored  energy  w.  To  produce  it, 
power,  p,  must  therefore  be  supplied  by  the  circuit.  Since  power 
is  current  times  voltage, 

p  =  e'i.  (2) 


12         ELECTRIC  DISCHARGES,  WAVES  AND  IMPULSES. 

To  produce  the  magnetic  field  $  of  the  current  i,  a  voltage  ef 
must  be  consumed  in  the  circuit,  which  with  the  current  i  gives 
the  power  p,  which  supplies  the  stored  energy  w  of  the  magnetic 
field  <i>.  This  voltage  er  is  called  the  inductance  voltage,  or  voltage 
consumed  by  self-induction. 

Since  no  power  is  required  to  maintain  the  field,  but  power  is 
required  to  produce  it,  the  inductance  voltage  must  be  propor- 
tional to  the  increase  of  the  magnetic  field: 

:'     ;  (3) 

or  by  (1), 

(4) 
If  i  and  therefore  $  decrease,  -r  and  therefore  e'  are  negative; 

that  is,  p  becomes  negative,  and  power  is  returned  into  the  circuit. 
The  energy  supplied  by  the  power  p  is 

w  =    I  p  dt, 
or  by  (2)  and  (4), 

w  =    I  Li  di; 

hence 

L*  (^ 

w  =  T  (5) 

is  the  energy  of  the  magnetic  field 

$  =  Li 

of  the  circuit. 

9.   Exactly  analogous  relations  exist  in  the  dielectric  field. 

The  dielectric  field,  or  dielectric  flux,  ty}  is  proportional  to  the 
voltage  6,  with  a  proportionality  factor,  C,  which  is  called  the 
capacity  of  the  circuit: 

f  =  Ce.  (6) 

The  dielectric  field  represents  stored  energy,  w.  To  produce  it, 
power,  p,  must,  therefore,  be  supplied  by  the  circuit.  Since  power 
is  current  times  voltage, 

p  =  i'e.  (7) 

To  produce  the  dielectric  field  ty  of  the  voltage  e,  a  current  ir 
must  be  consumed  in  the  circuit,  which  with  the  voltage  e  gives 


THE  ELECTRIC  FIELD.  13 

the  power  p,  which  supplies  the  stored  energy  w  of  the  dielectric 
field  ^.  This  current  i'  is  called  the  capacity  current,  or,  wrongly, 
charging  current  or  condenser  current. 

Since  no  power  is  required  to  maintain  the  field,  but  power  is 
required  to  produce  it,  the  capacity  current  must  be  proportional 
to  the  increase  of  the  dielectric  field: 


or  by  (6), 

i'  =  C^.  (9) 

de 
If  e  and  therefore  ^  decrease,  -j-  and  therefore  if  are  negative; 

that  is,  p  becomes  negative,  and  power  is  returned  into  the  circuit. 
The  energy  supplied  by  the  power  p  is 

w=j*pdt,  (10) 

or  by  (7)  and  (9), 

w  =  I  Cede; 
hence 

rw 

«  =  £  (ID 

is  the  energy  of  the  dielectric  field 

t  =  Ce 
of  the  circuit. 

As  seen,  the  capacity  current  is  the  exact  analogy,  with  regard 
to  the  dielectric  field,  of  the  inductance  voltage  with  regard  to  the 
magnetic  field;  the  representations  in  the  electric  circuit,  of  the 
energy  storage  in  the  field. 

The  dielectric  field  of  the  circuit  thus  is  treated  and  represented 
in  the  same  manner,  and  with  the  same  simplicity  and  perspicuity, 
as  the  magnetic  field,  by  using  the  same  conception  of  lines  of 
force. 

Unfortunately,  to  a  large  extent  in  dealing  with  the  dielectric 
fields  the  prehistoric  conception  of  the  electrostatic  charge  on  the 
conductor  still  exists,  and  by  its  use  destroys  the  analogy  between 
the  two  components  of  the  electric  field,  the  magnetic  and  the 


14         ELECTRIC  DISCHARGES,   WAVES  AND  IMPULSES. 

dielectric,  and  makes  the  consideration  of  dielectric  fields  un- 
necessarily complicated. 

There  obviously  is  no  more  sense  in  thinking  of  the  capacity 
current  as  current  which  charges  the  conductor  with  a  quantity 
of  electricity,  than  there  is  of  speaking  of  the  inductance  voltage 
as  charging  the  conductor  with  a  quantity  of  magnetism.  But 
while  the  latter  conception,  together  with  the  notion  of  a  quantity 
of  magnetism,  etc.,  has  vanished  since  Faraday's  representation 
of  the  magnetic  field  by  the  lines  of  magnetic  force,  the  termi- 
nology of  electrostatics  of  many  textbooks  still  speaks  of  electric 
charges  on  the  conductor,  and  the  energy  stored  by  them,  without 
considering  that  the  dielectric  energy  is  not  on  the  surface  of  the 
conductor,  but  in  the  space  outside  of  the  conductor,  just  as  the 
magnetic  energy. 

10.  All  the  lines  of  magnetic  force  are  closed  upon  themselves, 
all  the  lines  of  dielectric  force  terminate  at  conductors,  as  seen  in 
Fig.  8,  and  the  magnetic  field  and  the  dielectric  field  thus  can  be 
considered  as  a  magnetic  circuit  and  a  dielectric  circuit. 

To  produce  a  magnetic  flux  <£,  a  magnetomotive  force  F  is  required. 
Since  the  magnetic  field  is  due  to  the  current,  and  is  proportional 
to  the  current,  or,  in  a  coiled  circuit,  to  the  current  times  the  num- 
ber of  turns,  magnetomotive  force  is  expressed  in  current  turns  or 
ampere  turns. 

F  =  ni.  (12) 

If  F  is  the  m.m.f.,  I  the  length  of  the  magnetic  circuit,  energized 
by  F,  ,£ 

/  =  7  (13) 

is  called  the  magnetizing  force,  and  is  expressed  in  ampere  turns  per 
cm.  (or  industrially  sometimes  in  ampere  turns  per  inch). 

In  empty  space,  and  therefore  also,  with  very  close  approxi- 
mation, in  all  nonmagnetic  material,  /  ampere  turns  per  cm.  length 
of  magnetic  circuit  produce  3C  =  4  TT/  10"1  lines  of  magnetic  force 
per  square  cm.  section  of  the  magnetic  circuit.  (Here  the  factor 
10"1  results  from  the  ampere  being  10"1  of  the  absolute  or  cgs. 
unit  of  current.) 

(14) 


*  The  factor  4  *•  is  a  survival  of  the  original  definition  of  the  magnetic  field 
intensity  from  the  conception  of  the  magnetic  mass,  since  unit  magnetic  mass 
was  defined  as  that  quantity  of  magnetism  which  acts  on  an  equal  quantity  at 


THE  ELECTRIC  FIELD.  15 

is  called  the  magnetic-field  intensity.  It  is  the  magnetic  density, 
that  is,  the  number  of  lines  of  magnetic  force  per  cm2,  produced 
by  the  magnetizing  force  of  /  ampere  turns  per  cm.  in  empty  space. 
The  magnetic  density,  in  lines  of  magnetic  force  per  cm2,  pro- 
duced by  the  field  intensity  3C  in  any  material  is 

&  =  /z3C,  (15) 

where  ju  is  a  constant  of  the  material,  a  "  magnetic  conductivity," 
and  is  called  the  permeability.  ^  =  1  or  very  nearly  so  for  most 
materials,  with  the  exception  of  very  few,  the  so-called  magnetic 
materials:  iron,  cobalt,  nickel,  oxygen,  and  some  alloys  and  oxides 
of  iron,  manganese,  and  chromium. 

If  then  A  is  the  section  of  the  magnetic  circuit,  the  total  magnetic 
flux  is 

$  =  A®.  (16) 

Obviously,  if  the  magnetic  field  is  not  uniform,  equations  (13) 
and  (16)  would  be  correspondingly  modified;  /  in  (13)  would  be 
the  average  magnetizing  force,  while  the  actual  magnetizing  force 
would  vary,  being  higher  at  the  denser,  and  lower  at  the  less  dense, 
parts  of  the  magnetic  circuit: 

'-"• 

In  (16),  the  magnetic  flux  $  would  be  derived  by  integrating  the 
densities  (B  over  the  total  section  of  the  magnetic  circuit. 

ii.   Entirely  analogous  relations  exist  in  the- dielectric  circuit. 

To  produce  a  dielectric  flux  ^,  an  electromotive  force  e  is  required, 
which  is  measured  in  volts.  The  e.m.f.  per  unit  length  of  the 
dielectric  circuit  then  is  called  the  electrifying  force  or  the  voltage 
gradient,  and  is 

G  =  f-  (18)- 

unit  distance  with  unit  force.  The  unit  field  intensity,  then,  was  defined  as 
the  field  intensity  at  unit  distance  from  unit  magnetic  mass,  and  represented 
by  one  line  (or  rather  "tube")  of  magnetic  force.  The  magnetic  flux  of  unit 
magnetic  mass  (or  "unit  magnet  pole")  hereby  became  4w  lines  of  force,  and 
this  introduced  the  factor  4  TT  into  many  magnetic  quantities.  An  attempt 
to  drop  this  factor  4  TT  has  failed,  as  the  magnetic  units  were  already  too  well 
established. 

The  factor  1Q-1  also  appears  undesirable,  but  when  the  electrical  units 
were  introduced  the  absolute  unit  appeared  as  too  large  a  value  of  current  as 
practical  unit,  and  one-tenth  of  it  was  chosen  as  unit,  and  called  "ampere." 


16         ELECTRIC  DISCHARGES,   WAVES  AND  IMPULSES. 

This  gives  the  average  voltage  gradient,  while  the  actual  gradient 
in  an  ummiform  field,  as  that  between  two  conductors,  varies, 
being  higher  at  the  denser,  and  lower  at  the  less  dense,  portion  of 
the  field,  and  is 


then  is  the  dielectric-field  intensity,  and 

D  =  KK  (20) 

would  be  the  dielectric  density,  where  K  is  a  constant  of  the  material, 
the  electrostatic  or  dielectric  conductivity,  and  is  called  the  spe- 
cific capacity  or  permittivity. 

For  empty  space,  and  thus  with  close  approximation  for  air  and 
other  gases, 

1 

K    —    ~9» 

VL 

where 

v  =  3  X  1010 

is  the  velocity  of  light. 

It  is  customary,  however,  and  convenient,  to  use  the  permit- 
tivity of  empty  space  as  unity:  K  =  1.     This  changes  the  unit  of 

dielectric-field  intensity  by  the  factor  —  ,  and  gives:  dielectric-field 
intensity, 

dielectric  density, 


=  T^-oJ  (21) 

4  Try2 


D  =  KK,  (22) 

where  K  =  1  for  empty  space,  and  between  2  and  6  for  most  solids 
and  liquids,  rarely  increasing  beyond  6. 
The  dielectric  flux  then  is 

^  =  AD.  (23) 

12.  As  seen,  the  dielectric  and  the  magnetic  fields  are  entirely 
analogous,  and  the  corresponding  values  are  tabulated  in  the 
following  Table  I. 

*  The  factor  4  TT  appears  here  in  the  denominator  as  the  result  of  the  factor 
4*-  in  the  magnetic-field  intensity  5C,  due  to  the  relations  between  these 
quantities. 


THE  ELECTRIC  FIELD. 
TABLE  I. 


17 


Magnetic  Field. 

Dielectric  Field. 

Magnetic  flux: 
4>  =  Li  108  lines  of  magnetic  force. 

Dielectric  flux: 
^  =  Ce  lines  of  dielectric  force. 

Inductance  voltage: 

e'^n-^.  1Q-8  =  L  -jj  volts. 
at                  at 

Capacity  current: 
.,  _  d^  _  „  di 
dt          dt 

Magnetic  energy: 

Li2.     . 
w  =  -n-  joules. 

Dielectric  energy: 

Ce2 
w  =  -=-  joules. 

Magnetomotive  force: 
F  =  ni  ampere  turns. 

Electromotive  force: 
e  =  volts. 

Magnetizing  force: 

F 

f  =  -r  ampere  turns  per  cm. 

Electrifying  force  or  voltage  gra- 
dient: 

a 

G  =  j  volts  per  cm. 

Magnetic-field  intensity: 
3C  =  47r/10-1  lines   of  magnetic 
force  per  cm2. 

Dielectric-field  intensity: 

K  =  -  —  -  lines  of  dielectric  force 
4  Try2 

per  cm2. 

Magnetic  density: 
(B  =  M5C  lines  of  magnetic  force 
per  cm2. 

Dielectric  density: 
D  =  nK  lines  of  dielectric  force 
per  cm2. 

Permeability:  /* 

Permittivity  or  specific  capacity:  K 

Magnetic  flux: 
$  =  A($>  lines  of  magnetic  force. 

Dielectric  flux: 
^  =  AD  lines  of  dielectric  force. 

v  =  3  X  10  10  =  velocity  of  light. 

The  powers  of  10,  which  appear  in  some  expressions,  are  reduc- 
tion factors  between  the  absolute  or  cgs.  units  which  are  used  for 
$,  3C,  CB,  and  the  practical  electrical  units,  and  used  for  other 
constants. 

As  the  magnetic  field  and  the  dielectric  field  also  can  be  con- 
sidered as  the  magnetic  circuit  and  the  dielectric  circuit,  some 
analogy  exists  between  them  and  the  electric  circuit,  and  in  Table 
II  the  corresponding  terms  of  the  magnetic  circuit,  the  dielectric 
circuit,  and  the  electric  circuit  are  given. 


18        ELECTRIC  DISCHARGES,   WAVES  AND  IMPULSES. 

TABLE  II. 


Magnetic  Circuit. 

Dielectric  Circuit. 

Electric  Circuit. 

Magnetic  flux  (magnetic 

Dielectric  flux  (dielectric 

Electric  current: 

current): 

current)  : 

<£  =  lines   of  magnetic 

^  =  lines   of  dielectric 

i  =  electric   cur- 

force. 

force. 

rent. 

Magnetomotive  force: 

Electromotive  force: 

Voltage: 

F  =  ni  ampere  turns. 

e  =  volts. 

e  =  volts. 

Permeance: 

M  =  4?F 

Permittance  or  capacity: 

Conductance: 

Inductance: 

4irV2f  , 

i 

Q  —  -  mnos. 

~~F~  '        ~T 

henry. 

Reluctance: 

(Elastance  ?): 

Resistance: 

F 

1           e 

e 

& 

C      4*v*l>- 

T  ~~"  T  OIIIXIS. 

Magnetic  energy: 

Dielectric  energy: 

Electric  power: 

w=—  =  —  !Q-*  joules. 

Ce2      e^  .     , 
w  =  -JT-  =  -jr-  joules. 

p  =  ri2  =  ge2  —  ei 
watts. 

Magnetic  density: 

Dielectric  density: 

Electric-current 

density: 

(B  =  -j  =/z  JClinespercm2. 

A. 

D  =  ^  =  K/ninespercm2. 

7  =  -j  =  yG  am- 

A. 

perespercm2. 

Magnetizing  force: 

Dielectric  gradient: 

Electric  gradient: 

F 
/  =  j  ampere  turns  per 

G  =  j  volts  per  cm. 

G  =j  volts  per  cm. 

cm. 

Magnetic-field  intensity: 

Dielectric-field      inten- 

sity: 

OC  =  AT/. 

j^  ""   . 

Permeability: 

Permittivity   or   specific 

Conductivity: 

capacity: 

*-|- 

K_D 

y  =  ~  mho  —  cm. 

Cr 

Reluctivity: 

(Elastivity  ?): 

Resistivity: 

p  =  & 

1    .  K* 

1    G  , 
p  =  -  =  -?ohm  —  cm. 
y     I 

Specific  magnetic  energy: 

Specific  dielectric  energy: 

Specific  power: 

Awf2      /(B1A_8 

KGZ       GD  .     , 

Po  =  p/2  =  G2  =  GI 

W^O  ~~*         ^         —     c\    ^-^ 

WQ  —  -.  ;  —  ~~^:  —  JOUieS 

joules  per  cm3. 

4irV2          2     ' 

per  cm3. 

watts  per  cm3. 

LECTURE  III. 


SINGLE-ENERGY    TRANSIENTS    IN    CONTINUOUS- 
CURRENT   CIRCUITS. 

13.  The  simplest  electrical  transients  are  those  in  circuits  in 
which  energy  can  be  stored  in  one  form  only,  as  in  this  case  the 
change  of  stored  energy  can  consist  only  of  an  increase  or  decrease ; 
but  no  surge  or  oscillation  between  several  forms  of  energy  can 
exist.  Such  circuits  are  most  of  the  low-  and  medium-voltage 
circuits,  —  220  volts,  600  volts,  and  2200  volts.  In  them  the  capac- 
ity is  small,  due  to  the  limited  extent  of  the  circuit,  resulting  from 
the  low  voltage,  and  at  the  low  voltage  the  dielectric  energy  thus 
is  negligible,  that  is,  the  circuit  stores  appreciable  energy  only  by 
the  magnetic  field. 

A  circuit  of  considerable  capacity,  but  negligible  inductance,  if 
of  high  resistance,  would  also  give  one  form  of  energy  storage  only, 
in  the  dielectric  field.  The  usual  high-voltage  capacity  circuit,  as 
that  of  an  electrostatic  machine,  while  of  very  small  inductance, 
also  is  of  very  small  resistance,  and  the  momentary  discharge 
currents  may  be  very  consider- 
able, so  that  in  spite  of  the  very 

small    inductance,    considerable  __ 

magnetic-energy  storage  may  oc- 
cur; that  is,  the  system  is  one         eo 
storing  energy  in  two  forms,  and  ^ 

oscillations  appear,  as  in  the  dis-  '  ~ 

charge  of  the  Leyden  jar.  Fig  10._Magnetie  Single.energy 

Let,  as  represented  in  Fig.  10,  Transient, 

a  continuous  voltage  e0  be  im- 
pressed upon  a  wire  coil  of  resistance  r  and  inductance  L  (but 


negligible  capacity). 


A  current  iQ  =  —  flows  through  the  coil  and 

a  magnetic  field  $0  10~8  =  -  -  interlinks  with  the  coil.     Assuming 

now  that  the  voltage  e0  is  suddenly  withdrawn,  without  changing 

19 


20         ELECTRIC  DISCHARGES,   WAVES  AND  IMPULSES. 

the  constants  of  the  coil  circuit,  as  for  instance  by  short- 
circuiting  the  terminals  of  the  coil,  as  indicated  at  A,  with  no 
voltage  impressed  upon  the  coil,  and  thus  no  power  supplied  to  it, 
current  i  and  magnetic  flux  <£  of  the  coil  must  finally  be  zero. 
However,  since  the  magnetic  flux  represents  stored  energy,  it 
cannot  instantly  vanish,  but  the  magnetic  flux  must  gradually 
decrease  from  its  initial  value  3>o,  by  the  dissipation  of  its  stored 
energy  in  the  resistance  of  the  coil  circuit  as  i~r.  Plotting,  there- 
fore, the  magnetic  flux  of  the  coil  as  function  of  the  time,  in  Fig. 
11  A,  the  flux  is  constant  and  denoted  by  $0  up  to  the  moment  of 


Fig.  11.  —  Characteristics  of  Magnetic  Single-energy  Transient. 


time  where  the  short  circuit  is  applied,  as  indicated  by  the  dotted 
line  t0.  From  t0  on  the  magnetic  flux  decreases,  as  shown  by  curve 
<£.  Since  the  magnetic  flux  is  proportional  to  the  current,  the 
latter  must  follow  a  curve  proportional  to  <£,  as  shown  in  Fig.  IIB. 
The  impressed  voltage  is  shown  in  Fig.  1 1C  as  a  dotted  line;  it  is 
CQ  up  to  t0,  and  drops  to  0  at  t0.  However,  since  after  t0  a  current 
i  flows,  an  e.m.f.  must  exist  in  the  circuit,  proportional  to  the 
current. 

e  =  ri. 


SINGLE-ENERGY  TRANSIENTS.  21 

This  is  the  e.m.f.  induced  by  the  decrease  of  magnetic  flux  <£,  and 
is  therefore  proportional  to  the  rate  of  decrease  of  <£,  that  is,  to 

d<& 

-j- .     In  the  first  moment  of  short  circuit,  the  magnetic  flux  $  still 

has  full  value  3>0,  and  the  current  i  thus  also  full  value  iQ.  Hence, 
at  the  first  moment  of  short  circuit,  the  induced  e.m.f.  e  must  be 
equal  to  eQ,  that  is,  the  magnetic  flux  $  must  begin  to  decrease  at 
such  rate  as  to  induce  full  voltage  e0,  as  shown  in  Fig.  11C. 

The  three  curves  <£,  i,  and  e  are  proportional  to  each  other,  and 
as  e  is  proportional  to  the  rate  of  change  of  3>,  <£  must  be  propor- 
tional to  its  own  rate  of  change,  and  thus  also  i  and  e.  That  is, 
the  transients  of  magnetic  flux,  current,  and  voltage  follow  the 
law  of  proportionality,  hence  are  simple  exponential  functions,  as 
seen  in  Lecture  I: 


(1) 


<£,  i,  and  e  decrease  most  rapidly  at  first,  and  then  slower  and 
slower,  but  can  theoretically  never  become  zero,  though  prac- 
tically they  become  negligible  in  a  finite  time. 

The  voltage  e  is  induced  by  the  rate  of  change  of  the  magnetism, 
and  equals  the  decrease  of  the  number  of  lines  of  magnetic  force, 
divided  by  the  time  during  which  this  decrease  occurs,  multiplied 
by  the  number  of  turns  n  of  the  coil.  The  induced  voltage  e 
times  the  time  during  which  it  is  induced  thus  equals  n  times  the 
decrease  of  the  magnetic  flux,  and  the  total  induced  voltage, 
that  is,  the  area  of  the  induced-voltage  curve,  Fig.  11C,  thus 
equals  n  times  the  total  decrease  of  magnetic  flux,  that  is,  equals 
the  initial  current  i0  times  the  inductance  L: 


Zet  =  w£010-8  =  LiQ.  (2) 

Whatever,  therefore,  may  be  the  rate  of  decrease,  or  the  shape 
of  the  curves  of  $,  i,  and  e,  the  total  area  of  the  voltage  curve  must 
be  the  same,  and  equal  to  w£0  =  Li0. 

If  then  the  current  i  would  continue  to  decrease  at  its  initial 
rate,  as  shown  dotted  in  Fig.  115  (as  could  be  caused,  for  instance, 
by  a  gradual  increase  of  the  resistance  of  the  coil  circuit),  the 
induced  voltage  would  retain  its  initial  value  e0  up  to  the  moment 
of  time  t  =  tQ  +  T,  where  the  current  has  fallen  to  zero,  as 


22        ELECTRIC  DISCHARGES,   WAVES  AND  IMPULSES. 

shown  dotted  in  Fig.  11C.  The  area  of  this  new  voltage  curve 
would  be  e0T,  and  since  it  is  the  same  as  that  of  the  curve  e,  as 
seen  above,  it  follows  that  the  area  of  the  voltage  curve  e  is 


=  ri.r, 

and,  combining  (2)  and  (3),  i0  cancels,  and  we  get  the  value  of  T: 

:  •  .:'  :V  :  T-\-  >•'••;  •    (4) 

That  is,  the  initial  decrease  of  current,  and  therefore  of  mag- 
netic flux  and  of  induced  voltage,  is  such  that  if  the  decrease 
continued  at  the  same  rate,  the  current,  flux,  and  voltage  would 

become  zero  after  the  time  T  =  —  • 

r 

The  total  induced  voltage,  that  is,  voltage  times  time,  and 
therefore  also  the  total  current  and  magnetic  flux  during  the 
transient,  are  such  that,  when  maintained  at  their  initial  value, 

they  would  last  for  the  time  T  =  -=•  • 

Since  the  curves  of  current  and  voltage  theoretically  never 
become  zero,  to  get  an  estimate  of  the  duration  of  the  transient 
we  may  determine  the  time  in  which  the  transient  decreases  to 
half,  or  to  one-tenth,  etc.,  of  its  initial  value.  It  is  preferable, 
however,  to  estimate  the  duration  of  the  transient  by  the  time  T, 
which  it  would  last  if  maintained  at  its  initial  value.  That  is, 

the  duration  of  a  transient  is  considered  as  the  time  T  =  -  • 

r 

This  time  T  has  frequently  been  called  the  "  time  constant  " 
of  the  circuit. 

The  higher  the  inductance  L,  the  longer  the  transient  lasts, 
obviously,  since  the  stored  energy  which  the  transient  dissipates 
is  proportional  to  L. 

The  higher  the  resistance  r,  the  shorter  is  the  duration  of  the 
transient,  since  in  the  higher  resistance  the  stored  energy  is  more 
rapidly  dissipated. 

Using  the  time  constant  T  =  -  as  unit  of  length  for  the  abscissa, 

and  the  initial  value  as  unit  of  the  ordinates,  all  exponential 
transients  have  the  same  shape,  and  can  thereby  be  constructed 


SINGLE-ENERGY   TRANSIENTS. 


by  the  numerical  values  of  the  exponential  function,  y  =  e 
given  in  Table  III. 

TABLE  III. 

Exponential  Transient  of  Initial  Value  1  and  Duration  1. 
y  =  e~x.  e  =  2.71828. 


X 

y 

X 

y 

0 

1.000 

1.0 

0.368 

0.05 

0.951 

1.2 

0.301 

0.1 

0.905 

1.4 

0.247 

0.15 

0.860 

1.6 

0.202 

0.2 

0.819 

1.8 

0.165 

0.25 

0.779 

2.0 

0.135 

0.3 

0.741 

2.5 

0.082 

0.35 

0.705 

3.0 

0.050 

0.4 

0.670 

3.5 

0.030 

0.45 

0.638 

4.0 

0.018 

0.5 

0.607 

4.5 

0.011 

0.6 

0.549 

5.0 

0.007 

0.7 

0.497 

6.0 

0.002 

0.8 

0.449 

7.0 

0.001 

0.9 

0.407 

8.0 

0.000 

1.0 

0.368 

As  seen  in  Lecture  I,  the  coefficient  of  the  exponent  of  the 
single-energy  transient,  c,  is  equal  to  ^,  where  T  is  the  projection 
of  the  tangent  at  the  starting  moment  of  the  transient,  as  shown 
in  Fig.  11,  and  by  equation  (4)  was  found  equal  to  -.  That  is, 


r 

r 


and  the  equations  of  the  single-energy  magnetic  transient,  (1), 
thus  may  be  written  in  the  forms: 


I  =  lot~  c  (t  ~  'o)    =  IQ€ 

e  =  e0e~c('~'o)  =  e0e 


-  7  #  -  to 

=  ^Qe    L         , 

-  y  (t  -  *0) 

^r   /?„£      L 


(5) 


Usually,  the  starting  moment  of  the  transient  is  chosen  as  the 
zero  of  time,  Zo  =  0,  and  equations  (5)  then  assume  the  simpler 
form: 


24         ELECTRIC  DISCHARGES,   WAVES  AND  IMPULSES, 


(6) 


The  same  equations  may  be  derived  directly  by  the  integration 
of  the  differential  equation: 


where  L  -=-  is  the  inductance  voltage,  ri  the  resistance  voltage. 

and  their  sum  equals  zero,  as  the  coil  is  short-circuited. 
Equation  (7)  transposed  gives 


hence 


logi  =- 


i  =  Ce~~L\ 


and,  as  for  t  =  0:  i  =  to,  it  is:  C  =  i0;  hence 


14.  Usually  single-energy  transients  last  an  appreciable  time, 
and  thereby  become  of  engineering  importance  only  in  highly 
inductive  circuits,  as  motor  fields,  magnets,  etc. 

To  get  an  idea  on  the  duration  of  such  magnetic  transients, 
consider  a  motor  field: 

A  4-polar  motor  has  8  ml.  (megalines)  of  magnetic  flux  per 
pole,  produced  by  6000  ampere  turns  m.m.f.  per  pole,  and  dissi- 
pates normally  500  watts  in  the  field  excitation. 

That  is,  if  IQ  =  field-exciting  current,  n  =  number  of  field  turns 
per  pole,  r  =  resistance,  and  L  =  inductance  of  the  field-exciting 

circuit,  it  is 

iQ2r  =  500, 

hence 

500 


SINGLE-ENERGY  TRANSIENTS.  25 

The  magnetic  flux  is  $0  =  8  X  106,  and  with  4  n  total  turns 
the  total  number  of  magnetic  interlinkages  thus  is 

4  n$0  =  32  n  X  106, 
hence  the  inductance 

L0~8      .32  n 


T 
L  = 


^o 


, 
henrys. 


The  field  excitation  is 

ra'o  =  6000  ampere  turns, 

6000 


hence 
hence 
and 


n  = 


,       .32  X  6000  , 

L  =  -  —r henrys, 


*<r 
L      1920 


0  OA 
-  3'84  sec' 


That  is,  the  stored  magnetic  energy  could  maintain  full  field 
excitation  for  nearly  4  seconds. 

It  is  interesting  to  note  that  the  duration  of  the  field  discharge 
does  not  depend  on  the  voltage,  current,  or  size  of  the  machine, 
but  merely  on,  first,  the  magnetic  flux  and  m.m.f.,  —  which 
determine  the  stored  magnetic  energy,  —  and,  second,  on  the 
excitation  power,  which  determines  the  rate  of  energy  dissipation. 

15.  Assume  now  that  in  the  moment  where  the  transient  be- 
gins the  resistance  of  the  coil  in  Fig.  10  is  increased,  that  is,  the 


I 


Fig.  12.  —  Magnetic  Single-energy  Transient. 

coil  is  not  short-circuited '  upon  itself,  but  its  circuit  closed  by  a 
resistance  r1 '.  Such  would,  for  instance,  be  the  case  in  Fig.  12, 
when  opening  the  switch  S. 


26         ELECTRIC  DISCHARGES,   WAVES  AND  IMPULSES. 

The  transients  of  magnetic  flux,  current,  and  voltage  are  shown 
as  A,  B,  and  C  in  Fig.  13. 

The  magnetic  flux  and  therewith  the  current  decrease  from  the 
initial  values  $o  and  i0  at  the  moment  to  of  opening  the  switch  S, 
on  curves  which  must  be  steeper  than  those  in  Fig.  11,  since  the 
current  passes  through  a  greater  resistance,  r  +  r',  and  thereby 
dissipates  the  stored  magnetic  energy  at  a  greater  rate. 


Fig.  13.  —  Characteristics  of  Magnetic  Single-energy  Transient. 

The  impressed  voltage  eQ  is  withdrawn  at  the  moment  t0,  and  a 
voltage  thus  induced  from  this  moment  onward,  of  such  value  as 
to  produce  the  current  i  through  the  resistance  r  +  r'.  In  the 
first  moment,  to,  the  current  is  still  iQ,  and  the  induced  voltage 
thus  must  be 

eo   =  io  (r  +  r'), 

while  the  impressed  voltage,  before  to,  was 

eQ  =  ior; 

hence  the  induced  voltage  eo'  is  greater  than  the  impressed  volt- 
age eo,  in  the  same  ratio  as  the  resistance  of  the  discharge  circuit 
r  +  r'  is  greater  than  the  resistance  of  the  coil  r  through  which  the 
impressed  voltage  sends  the  current 


e0 


SINGLE-ENERGY  TRANSIENTS.  27 

The  duration  of  the  transient  now  is 

T          L 

T  =  7+-r" 

that  is,  shorter  in  the  same  proportion  as  the  resistance,  and 
thereby  the  induced  voltage  is  higher. 

If  rf  =  oo ,  that  is,  no  resistance  is  in  shunt  to  the  coil,  but  the 
circuit  is  simply  opened,  if  the  opening  were  instantaneous,  it 
would  be :  e0f  =  co  ;  that  is,  an  infinite  voltage  would  be  induced. 
That  is,  the  insulation  of  the  coil  would  be  punctured  and  the 
circuit  closed  in  this  manner. 

The  more  rapid,  thus,  the  opening  of  an  inductive  circuit,  the 
higher  is  the  induced  voltage,  and  the  greater  the  danger  of  break- 
down. Hence  it  is  not  safe  to  have  too  rapid  circuit-opening 
devices  on  inductive  circuits. 

To  some  extent  the  circuit  protects  itself  by  an  arc  following  the 
blades  of  the  circuit-opening  switch,  and  thereby  retarding  the  cir- 
cuit opening.  The  more  rapid  the  mechanical  opening  of  the 
switch,  the  higher  the  induced  voltage,  and  further,  therefore,  the 
arc  follows  the  switch  blades  and  maintains  the  circuit. 

16.  Similar  transients  as  discussed  above  occur  when  closing  a 
circuit  upon  an  impressed  voltage,  or  changing  the  voltage,  or  the 
current,  or  the  resistance  or  inductance  of  the  circuit.  A  discus- 
sion of  the  infinite  variety  of  possible  combinations  obviously 
would  be  impossible.  However,  they  can  all  be  reduced  to  the 
same  simple  case  discussed  above,  by  considering  that  several 
currents,  voltages,  magnetic  fluxes,  etc.,  in  the  same  circuit  add 
algebraically,  without  interfering  with  each  other  (assuming,  as 
done  here,  that  magnetic  saturation  is  not  approached). 

If  an  e.m.f.  e\  produces  a  current  i\  in  a  circuit,  and  an  e.m.f.  ez 
produces  in  the  same  circuit  a  current  i2,  then  the  e.m.f.  e\  +  ez 
produces  the  current  i\  -\-  1%,  as  is  obvious. 

If  now  the  voltage  e\  +  ez,  and  thus  also  the  current  ii  +  iZj  con- 
sists of  a  permanent  term,  e\  and  ii,  and  a  transient  term,  e2  and  iz, 
the  transient  terms  ez,  iz  follow  the  same  curves,  when  combined 
with  the  permanent  terms  e\,  i\,  as  they  would  when  alone  in  the 
circuit  (the  case  above  discussed).  Thus,  the  preceding  discus- 
sion applies  to  all  magnetic  transients,  by  separating  the  transient 
from  the  permanent  term,  investigating  it  separately,  and  then 
adding  it  to  the  permanent  term. 


28         ELECTRIC  DISCHARGES,   WAVES  AND  IMPULSES. 


The  same  reasoning  also  applies  to  the  transient  resulting  from 
several  forms  of  energy  storage  (provided  that  the  law  of  propor- 
tionality of  i,  e,  $,  etc.,  applies),  and  makes  it  possible,  in  inves- 
tigating the  phenomena  during  the  transition  period  of  energy 
readjustment,  to  separate  the  permanent  and  the  transient  term, 
and  discuss  them  separately. 


B 


Fig.  14.  —  Single-energy  Starting  Transient  of  Magnetic  Circuit. 

For  instance,  in  the  coil  shown  in  Fig.  10,  let  the  short  circuit  A 
be  opened,  that  is,  the  voltage  eQ  be  impressed  upon  the  coil.  At 
the  moment  of  time,  tQ,  when  this  is  done,  current  i,  magnetic 
flux  <£,  and  voltage  e  on  the  coil  are  zero.  In  final  condition,  after 
the  transient  has  passed,  the  values  i0, 3>0,  e0  are  reached.  We  may 
then,  as  discussed  above,  separate  the  transient  from  the  perma- 
nent term,  and  consider  that  at  the  time  U  the  coil  has  a  permanent 
current  i0,  permanent  flux  <i>0,  permanent  voltage  e0,  and  in  addi- 


SINGLE-ENERGY   TRANSIENTS.  29 

tion  thereto  a  transient  current  —i0j  a  transient  flux  —  <£0,  and  a 
transient  voltage  —  eQ.  These  transients  are  the  same  as  in  Fig.  11 
(only  with  reversed  direction).  Thus  the  same  curves  result,  and 
to  them  are  added  the  permanent  values  i0,  <J>0,  e0.  This  is  shown 
in  Fig.  14. 

A  shows  the  permanent  flux  <£0,  and  the  transient  flux  —  <J>0, 
which  are  assumed,  up  to  the  time  tQ,  to  give  the  resultant  zero 
flux.  The  transient  flux  dies  out  by  the  curve  <£',  in  accordance 
with  Fig.  11.  &  added  to  <£0  gives  the  curve  3>,  which  is  the  tran- 
sient from  zero  flux  to  the  permanent  flux  3>0. 

In  the  same  manner  B  shows  the  construction  of  the  actual 
current  change  i  by  the  addition  of  the  permanent  current  iQ  and 
the  transient  current  i',  which  starts  from  —iQ  at  to. 

C  then  shows  the  voltage  relation:  eQ  the  permanent  voltage,  e' 
the  transient  voltage  which  starts  from  —  e0  at  t0,  and  e  the  re- 
sultant or  effective  voltage  in  the  coil,  derived  by  adding  eQ  and  e'. 


LECTURE   IV. 

SINGLE-ENERGY  TRANSIENTS   IN   ALTERNATING- 
CURRENT  CIRCUITS. 

17.  Whenever  the  conditions  of  an  electric  circuit  are  changed 
in  such  a  manner  as  to  require  a  change  of  stored  energy,  a  transi- 
tion period  appears,  during  which  the  stored  energy  adjusts  itself 
from  the  condition  existing  before  the  change  to  the  condition 
after  the  change.  The  currents  in  the  circuit  during  the  transition 
period  can  be  considered  as  consisting  of  the  superposition  of 
the  permanent  current,  corresponding  to  the  conditions  after  the 
change,  and  a  transient  current,  which  connects  the  current  value 
before  the  change  with  that  brought  about  by  the  change.  That 
is,  if  i\  =  current  existing  in  the  circuit  immediately  before,  and 
thus  at  the  moment  of  the  change  of  circuit  condition,  and  i%  = 
current  which  should  exist  at  the  moment  of  change  in  accordance 
with  the  circuit  condition  after  the  change,  then  the  actual  current 
ii  can  be  considered  as  consisting  of  a  part  or  component  iz,  and  a 
component  ii  —  iz  —  IQ.  The  former,  iz,  is  permanent,  as  result- 
ing from  the  established  circuit  condition.  The  current  compo- 
nent IQ,  however,  is  not  produced  by  any  power  supply,  but  is  a 
remnant  of  the  previous  circuit  condition,  that  is,  a  transient,  and 
,  therefore  gradually  decreases  in  the  manner  as  discussed  in  para- 
graph 13,  that  is,  with  a  duration  T  =  —  - 

The  permanent  current  i2  may  be  continuous,  or  alternating,  or 
may  be  a  changing  current,  as  a  transient  of  long  duration,  etc. 

The  same  reasoning  applies  to  the  voltage,  magnetic  flux,  etc. 

Thus,  let,  in  an  alternating-current  circuit  traversed  by  current 
t'i,  in  Fig.  15  A,  the  conditions  be  changed,  at  the  moment  t  =  0, 
so  as  to  produce  the  current  i2.  The  instantaneous  value  of  the 
current  ii  at  the  moment  t  =  0  can  be  considered  as  consisting 
of  the  instantaneous  value  of  the  permanent  current  i2,  shown 
dotted,  and  the  transient  io  =  i\  —  i*.  The  latter  gradually  dies 

down,  with  the  duration  T  —  — ,  on  the  usual  exponential  tran- 

30 


SINGLE-ENERGY  TRANSIENTS. 


31 


sient,  shown  dotted  in  Fig.  15.  Adding  the  transient  current  iQ 
to  the  permanent  current  i2  gives  the  total  current  during  the 
transition  period,  which  is  shown  in  drawn  line  in  Fig.  15. 

As  seen,  the  transient  is  due  to  the  difference  between  the 
instantaneous  value  of  the  current  i\  which  exists,  and  that  of 
the  current  i2  which  should  exist  at  the  moment  of  change,  and 


Fig.  15.  —  Single-energy  Transient  of  Alternating-current  Circuit. 


thus  is  the  larger,  the  greater  the  difference  between  the  two 
currents,  the  previous  and  the  after  current.  It  thus  disappears 
if  the  change  occurs  at  the  moment  when  the  two  currents  ii 
and  12  are  equal,  as  shown  in  Fig.  15B,  and  is  a  maximum,  if  the 
change  occurs  at  the  moment  when  the  two  currents  i\  and  iz 
have  the  greatest  difference,  that  is,  at  a  point  one-quarter  period 
or  90  degrees  distant  from  the  intersection  of  i\  and  12,  as  shown 
in  Fig.  15C. 


32 


ELECTRIC  DISCHARGES,   WAVES  AND  IMPULSES. 


If  the  current  ii  is  zero,  we  get  the  starting  of  the  alternating 
current  in  an  inductive  circuit,  as  shown  in  Figs.  16,  A,  B,  C.  The 
starting  transient  is  zero,  if  the  circuit  is  closed  at  the  moment 
when  the  permanent  current  would  be  zero  (Fig.  16B),  and  is  a 
maximum  when  closing  the  circuit  at  the  maximum  point  of  the 
permanent-current  wave  (Fig.  16C).  The  permanent  current  and 
the  transient  components  are  shown  dotted  in  Fig.  16,  and  the 
resultant  or  actual  current  in  drawn  lines. 


B 


Fig.  16.  —  Single-energy  Starting  Transient  of  Alternating-current  Circuit. 

1 8.  Applying  the  preceding  to  the  starting  of  a  balanced 
three-phase  system,  we  see,  in  Fig.  17 A,  that  in  general  the  three 
transients  t'i°,  i20,  and  4°  of  the  three  three-phase  currents  ii,  iz,  is 
are  different,  and  thus  also  the  shape  of  the  three  resultant 
currents  during  the  transition  period.  Starting  at  the  moment 
of  zero  current  of  one  phase,  ii,  Fig.  175,  there  is  no  transient  for 
this  current,  while  the  transients  of  the  other  two  currents,  iz 
and  i3,  are  equal  and  opposite,  and  near  their  maximum  value. 
Starting,  in  Fig.  17C,  at  the  maximum  value  of  one  current  ia, 
we  have  the  maximum  value  of  transient  for  this  current  i'3°,  while 
the  transients  of  the  two  other  currents,  i\  and  ii,  are  equal,  have 


SINGLE-ENERGY  TRANSIENTS. 


33 


half  the  value  of  13°,  and  are  opposite  in  direction  thereto.  In 
any  case,  the  three  transients  must  be  distributed  on  both  sides 
of  the  zero  line.  This  is  obvious:  if  ii,  i2',  and  is'  are  the  instan- 
taneous values  of  the  permanent  three-phase  currents,  in  Fig. 
17,  the  initial  values  of  their  transients  are:  —i\,  —  iz,  —is- 


Fig.  17.  —  Single-energy  Starting  Transient  of  Three-phase  Circuit. 

Since  the  sum  of  the  three  three-phase  currents  at  every  moment 
is  zero,  the  sum  of  the  initial  values  of  the  three  transient  currents 
also  is  zero.  Since  the  three  transient  curves  ii°,  i'2°,  iz°  are  pro- 
portional to  each  other  fas  exponential  curves  of  the  same  dura- 
tion T  =  — ],  and  the  sum  of  their  initial  values  is  zero,  it  follows 


34         ELECTRIC  DISCHARGES,   WAVES  AND  IMPULSES. 

that  the  sum  of  their  instantaneous  values  must  be  zero  at  any 
moment,  and  therefore  the  sum  of  the  instantaneous  values  of 
the  resultant  currents  (shown  in  drawn  line)  must  be  zero  at  any 
moment,  not  only  during  the  permanent  condition,  but  also  dur- 
ing the  transition  period  existing  before  the  permanent  condi- 
tion is  reached. 

It  is  interesting  to  apply  this  to  the  resultant  magnetic  field 
produced  by  three  equal  three-phase  magnetizing  coils  placed 
under  equal  angles,  that  is,  to  the  starting  of  the  three-phase 
rotating  magnetic  field,  or  in  general  any  polyphase  rotating 
magnetic  field. 


Fig.  18.  —  Construction  of  Starting  Transient  of  Rotating  Field. 

As  is  well  known,  three  equal  magnetizing  coils,  placed  under 
equal  angles  and  excited  by  three-phase  currents,  produce  a  result- 
ant magnetic  field  which  is  constant  in  intensity,  but  revolves 
synchronously  in  space,  and  thus  can  be  represented  by  a  concen- 
tric circle  a,  Fig.  18. 

This,  however,  applies  only  to  the  permanent  condition.  In 
the  moment  of  start,  all  the  three  currents  are  zero,  and  their 
resultant  magnetic  field  thus  also  zero,  as  shown  above.  Since 
the  magnetic  field  represents  stored  energy  and  thus  cannot  be 
produced  instantly,  a  transient  must  appear  in  the  building  up  of 
the  rotating  field.  This  can  be  studied  by  considering  separately 


SINGLE-ENERGY   TRANSIENTS. 


35 


the  permanent  and  the  transient  components  of  the  three  currents, 
as  is  done  in  the  preceding.  Let  ii,  i2)  is  be  the  instantaneous 
values  of  the  permanent  currents  at  the  moment  of  closing  the 
circuit,  t  =  0.  Combined,  these  would  give  the  resultant  field 
(Mo  in  Fig.  18.  The  three  transient  currents  in  this  moment 
are  i'i°  =—ii,  i^_==  —  i2,  13°  =—i^',  and  combined  these  give  a 
resultant  field  OB0,  equal  and  opposite  to  OA0  in  Fig.  18.  The 
permanent  field  rotates  synchronously  on  the  concentric  circle  a; 
the  transient  field  OB  remains  constant  in  the  direction  OB0, 
since  all  three  transient  components  of  current  decrease  in  propor- 
tion to  each  other.  It  decreases,  however,  with  the  decrease  of 
the  transient  current,  that  is,  shrinks  together  on  the  line  BQ0. 
The  resultant  or  actual  field  thus  is  the  combination  of  the  per- 
manent fields,  shown  as  OAi  OA2,  .  .  .  ,  and  the  transient  fields, 
shown  as  OBi,  OBZ,  etc.,  and  derived  thereby  by  the  parallelo- 
gram law,  as  shown  in  Fig.  18,  as  OC\,  OC2,  etc.  In  this  diagram, 
Bid,  B2C2)  etc.,  are  equal  to  OAi,  OA2,  etc.,  that  is,  to  the  radius 
of  the  permanent  circle  a.  That  is,  while  the  rotating  field  in 
permanent  condition  is  represented  by  the  concentric  circle  a, 
the  resultant  field  during  the  transient  or  starting  period  is  repre- 
sented by  a  succession  of  arcs  of  circles  c,  the  centers  of  which 
move  from  BQ  in  the  moment  of  start,  on  the  line  BQ0  toward  0, 
and  can  be  constructed  hereby  by  drawing  from  the  successive 
points  B0,  BI}  B2)  which  correspond  to  successive  moments  of 
time  0,  tij  t2  ...  ,  radii  BiCi,  B2C2,  etc.,  under  the  angles,  that  is, 
in  the  direction  corresponding  to  the  time  0,  ^  t2,  etc.  This  is 
done  in  Fig.  19,  and  thereby  the  transient  of  the  rotating  field 
is  constructed. 


Fig.  19.  —  Starting  Transient  of  Rotating  Field:  Polar  Form. 


36        ELECTRIC  DISCHARGES,   WAVES  AND  IMPULSES. 

_From  this  polar  diagram  of  the  rotating  field,  in  Fig.  19,  values 
OC  can  now  be  taken,  corresponding  to  successive  moments  of 
time,  and  plotted  in  rectangular  coordinates,  as  done  in  Fig.  20. 
As  seen,  the  rotating  field  builds  up  from  zero  at  the  moment  of 
closing  the  circuit,  and  reaches  the  final  value  by  a  series  of  oscil- 
lations ;  that  is,  it  first  reaches  beyond  the  permanent  value,  then 
drops  below  it,  rises  again  beyond  it,  etc. 


3  4     cycles 


Fig.  20.  —  Starting  Transient  of  Rotating  Field:  Rectangular  Form. 

We  have  here  an  oscillatory  transient,  produced  in  a  system 
with  only  one  form  of  stored  energy  (magnetic  energy),  by  the 
combination  of  several  simple  exponential  transients.  How- 
ever, it  must  be  considered  that,  while  energy  can  be  stored 
in  one  form  only,  as  magnetic  energy,  it  can  be  stored  in  three 
electric  circuits,  and  a  transfer  of  stored  magnetic  energy  between 
the  three  electric  circuits,  and  therewith  a  surge,  thus  can 
occur. 

It  is  interesting  to  note  that  the  rot  at  ing-field  transient  is 
independent  of  the  point  of  the  wave  at  which  the  circuit  is 
closed.  That  is,  while  the  individual  transients  of  the  three 
three-phase  currents  vary  in  shape  with  the  point  of  the  wave  at 
which  they  start,  as  shown  in  Fig.  17,  their  polyphase  resultant 
always  has  the  same  oscillating  approach  to  a  uniform  rotating 

field,  of  duration  T  —  —  • 
r 

The  maximum  value,  which  the  magnetic  field  during  the  transi- 
tion period  can  reach,  is  limited  to  less  than  double  the  final  value, 
as  is  obvious  from  the  construction  of  the  'field,  Fig.  19.  It  is 
evident  herefrom,  however,  that  in  apparatus  containing  rotating 
fields,  as  induction  motors,  polyphase  synchronous  machines,  etc., 
the  resultant  field  may  under  transient  conditions  reach  nearly 
double  value,  and  if  then  it  reaches  far  above  magnetic  saturation, 
excessive  momentary  currents  may  appear,  similar  as  in  starting 
transformers  of  high  magnetic  density.  In  polyphase  rotary 


SINGLE-ENERGY  TRANSIENTS.  37 

apparatus,  however,  these  momentary  starting  currents  usually 
are  far  more  limited  than  in  transformers,  by  the  higher  stray  field 
(self-inductive  reactance),  etc.,  of  the  apparatus,  resulting  from 
the  air  gap  in  the  magnetic  circuit. 

19.  As  instance  of  the  use  of  the  single-energy  transient  in 
engineering  calculations  may  be  considered  the  investigation  of 
the  momentary  short-circuit  phenomena  of  synchronous  alter- 
nators. In  alternators,  especially  high-speed  high-power  mar 
chines  as  turboalternators,  the  momentary  short-circuit  current 
may  be  many  times  greater  than  the  final  or  permanent  short- 
circuit  current,  and  this  excess  current  usually  decreases  very 
slowly,  lasting  for  many  cycles.  At  the  same  time,  a  big  cur- 
rent rush  occurs  in  the  field.  This  excess  field  current  shows 
curious  pulsations,  of  single  and  of  double  frequency,  and  in 
the  beginning  the  armature  currents  also  show  unsymmetrical 
shapes.  Some  oscillograms  of  three-phase,  quarter-phase,  and 
single-phase  short  circuits  of  turboalternators  are  shown  in  Figs. 
25  to  28. 

By  considering  the  transients  of  energy  storage,  these  rather 
complex-appearing  phenomena  can  be  easily  understood,  and  pre- 
determined from  the  constants  of  the  machine  with  reasonable 
exactness. 

In  an  alternator,  the  voltage  under  load  is  affected  by  armature 
reaction  and  armature  self-induction.  Under  permanent  condi- 
tion, both  usually  act"  in  the  same  way,  reducing  the  voltage  at 
noninductive  and  still  much  more  at  inductive  load,  and  increasing 
it  at  antiinductive  load;  and  both  are  usually  combined  in  one 
quantity,  the  synchronous  reactance  XQ.  In  the  transients  result- 
ing from  circuit  changes,  as  short  circuits,  the  self-inductive 
armature  reactance  and  the  magnetic  armature  reaction  act  very 
differently:*  the  former  is  instantaneous  in  its  effect,  while  the 
latter  requires  time.  The  self-inductive  armature  reactance  Xi 
consumes  a  voltage  x\i  by  the  magnetic  flux  surrounding  the 
armature  conductors,  which  results  from  the  m.m.f .  of  the  armature 
current,  and  therefore  requires  a  component  of  the  magnetic-field 
flux  for  its  production.  As  the  magnetic  flux  and  the  current 
which  produces  it  must  be  simultaneous  (the  former  being  an 
integral  part  of  the  phenomenon  of  current  flow,  as  seen  in  Lecture 
II),  it  thus  follows  that  the  armature  reactance  appears  together 
*  So  also  in  their  effect  on  synchronous  operation,  in  hunting,  etc. 


38        ELECTRIC  DISCHARGES,   WAVES  AND  IMPULSES. 

with  the  armature  current,  that  is,  is  instantaneous.  The  arma- 
ture reaction,  however,  is  the  m.m.f.  of  the  armature  current  in  its 
reaction  on  the  m.m.f.  of  the  field-exciting  current.  That  is,  that 
part  xz  =  XQ  —  Xi  of  the  synchronous  reactance  which  corresponds 
to  the  armature  reaction  is  not  a  true  reactance  at  all,  consumes 
no  voltage,  but  represents  the  consumption  of  field  ampere  turns 
by  the  m.m.f.  of  the  armature  current  and  the  corresponding 
change  of  field  flux.  Since,  however,  the  field  flux  represents 
stored  magnetic  energy,  it  cannot  change  instantly,  and  the  arma- 
ture reaction  thus  does  not  appear  instantaneously  with  the  arma- 
ture current,  but  shows  a  transient  which  is  determined  essentially 
by  the  constants  of  the  field  circuit,  that  is,  is  the  counterpart  of 
the  field  transient  of  the  machine. 

If  then  an  alternator  is  short-circuited,  in  the  first  moment  only 
the  true  self -inductive  part  Xi  of  the  synchronous  reactance  exists, 

and  the  armature  current  thus  is  i\  =  — ,  where  e0  is  the  induced 

Xi 

e.m.f.,  that  is,  the  voltage  corresponding  to  the  magnetic-field 
excitation  flux  existing  before  the  short  circuit.  Gradually  the 
armature  reaction  lowers  the  field  flux,  in  the  manner  as  repre- 
sented by  the  synchronous  reactance  x0,  and  the  short-circuit  cur- 
rent decreases  to  the  value  i'0  =  —  • 

XQ 

The  ratio  of  the  momentary  short-circuit  current  to  the  perma- 
nent short-circuit  current  thus  is,  approximately,  the  ratio  —  =  —  > 

IQ        Xi 

that  is,  synchronous  reactance  to  self-inductive  reactance,  or 
armature  reaction  plus  armature  self-induction,  to  armature 
self-induction.  In  machines  of  relatively  low  self-induction 
and  high  armature  reaction,  the  momentary  short-circuit  cur- 
rent thus  may  be  many  times  the  permanent  short-circuit 
current. 

The  field  flux  remaining  at  short  circuit  is  that  giving  the  volt- 
age consumed  by  the  armature  self-induction,  while  the  decrease 
of  field  flux  between  open  circuit  and  short  circuit  corresponds  to 
the  armature  reaction.  The  ratio  of  the  open-circuit  field  flux  to 
the  short-circuit  field  flux  thus  is  the  ratio  of  armature  reaction 
plus  self-induction,  to  the  self-induction;  or  of  the  synchronous 

reactance  to  the  self-inductive  reactance:  —  • 


SINGLE-ENERGY  TRANSIENTS.  39 

Thus  it  is: 

momentary  short-circuit  current  _  open-circuit  field  flux  *  _ 
permanent  short-circuit  current  ~"   short-circuit  field  flux 
armature  reaction  plus  self-induction  _  synchronous  reactance  _  XQ 
self-induction  self-inductive  reactance  ~~  x\ 

20.  Let  $1  =  field  flux  of  a  three-phase  alternator  (or,  in  general, 
polyphase  alternator)  at  open  circuit,  and  this  alternator  be  short- 
circuited  at  the  time  t  =  0.  The  field  flux  then  gradually  dies 
down,  by  the  dissipation  of  its  energy  in  the  field  circuit,  to  the 
short-circuit  field  flux  3>0,  as  indicated  by  the  curve  $  in  Fig.  21A. 
If  m  =  ratio 

armature  reaction  plus  self-induction  _  XQ 
armature  self-induction  ~  x\ 

it  is  $1  =  m$0,  and  the  initial  value  of  the  field  flux  consists  of  the 
permanent  part  <i>0,  and  the  transient  part  <£'  =  $1  —  <£0  =  (m—l) 
$0.  This  is  a  rather  slow  transient,  frequently  of  a  duration  of  a 
second  or  more. 

The  armature  currents  i1}  ^  iz  are  proportional  to  the  field  flux 
$  which  produces  them,  and  thus  gradually  decrease,  from  initial 
values,  which  are  as  many  times  higher  than  the  final  values  as  $1 
is  higher  than  3>0,  or  m  times,  and  are  represented  in  Fig.  21 B. 

The  resultant  m.m.f.  of  the  armature  currents,  or  the  armature 
reaction,  is  proportional  to  the  currents,  and  thus  follows  the  same 
field  transient,  as  shown  by  F  in  Fig.  2 1C. 

The  field-exciting  current  is  i0  at  open  circuit  as  well  as  in  the 
permanent  condition  of  short  circuit.  In  the  permanent  condition 
of  short  circuit,  the  field  current  iQ  combines  with  the  armature 
reaction  F0,  which  is  demagnetizing,  to  a  resultant  m.m.f.,  which 
produces  the  short-circuit  flux  3>0.  During  the  transition  period 
the  field  flux  $  is  higher  than  3>0,  and  the  resultant  m.m.f.  must 
therefore  be  higher  in  the  same  proportion.  Since  it  is  the  dif- 
ference between  the  field  current  and  the  armature  reaction  F,  and 
the  latter  is  proportional  to  3>,  the  field  current  thus  must  also  be 

*  If  the  machine  were  open-circuited  before  the  short  circuit,  otherwise 
the  field  flux  existing  before  the  short  circuit.  It  herefrom  follows  that  the 
momentary  short-circuit  current  essentially  depends  on  the  field  flux,  and 
thereby  the  voltage  of  the  machine,  before  the  short  circuit,  but  is  practically 
independent  of  the  load  on  the  machine  before  the  short  circuit  and  the  field 
excitation  corresponding  to  this  load. 


40        ELECTRIC  DISCHARGES,   WAVES  AND  IMPULSES. 

proportional  to  $>.     Thus,  as  it  is  i  =  iQ  at  <£0,  during  the  transition 

<£ 
period  it  is  i  =  —  iQ.     Hence,  the  field-exciting  current  traverses 

<PO 

the  same  transient,  from  an  initial  value  iY  to  the  normal  value  i'0, 
as  the  field  flux  3>  and  the  armature  currents. 


B 


Fig.  21.  —  Construction  of  Momentary  Short  Circuit  Characteristic  of  Poly- 
phase Alternator. 

Thus,  at  the  moment  of  short  circuit  a  sudden  rise  of  field 
current  must  occur,  to  maintain  the  field  flux  at  the  initial  value 
$1  against  the  demagnetizing  armature  reaction.  In  other  words, 
the  field  flux  $  decreases  at  such  a  rate  as  to  induce  in  the  field 
circuit  the  e.m.f.  required  to  raise  the  field  current  in  the  propor- 
tion m,  from  iQ  to  i0f,  and  maintain  it  at  the  values  corresponding 
to  the  transient  i,  Fig.  2 ID. 

As  seen,  the  transients  3>;  z'i,  i'2,  iz]  F;  i  are  proportional  to  each 
other,  and  are  a  field  transient.  If  the  field,  excited  by  current  iQ 


SINGLE-ENERGY   TRANSIENTS.  41 

at  impressed  voltage  e0,  were  short-circuited  upon  itself,  in  the 
first  moment  the  current  in  the  field  would  still  be  iQ,  and  there- 
fore-the  voltage  e0  would  have  to  be  induced  by  the  decrease  of 
magnetic  flux ;  and  the  duration  of  the  field  transient,  as  discussed 

in  Lecture  III,  would  be  TQ  =  —  - 

ro 

The  field  current  in  Fig.  2  ID,  of  the  alternator  short-circuit 
transient,  starts  with  the  value  ij  =  mi0,  and  if  eQ  is  the  e.m.f. 
supplied  in  the  field-exciting  circuit  from  a  source  of  constant 
voltage  supply,  as  the  exciter,  to  produce  the  current  i0f,  the 
voltage  Co'  =  meo  must  be  acting  in  the  field-exciting  circuit;  that 
is,  in  addition  to  the  constant  exciter  voltage  e0,  a  voltage  (m  —  I)e0 
must  be  induced  in  the  field  circuit  by  the  transient  of  the  mag- 
netic flux.  As  a  transient  of  duration  —  induces  the  voltage  e0, 

TO 

to  induce  the  voltage  (m  —  I)e0  the  duration  of  the  transient  must 
be 


-1  o  —  /          -i  \      ) 
(m-  1)  TO 

where  L0  =  inductance,  r0  =  total  resistance  of  field-exciting  cir- 
cuit (inclusive  of  external  resistance). 

The  short-circuit  transient  of  an  alternator  thus  usually  is  of 
shorter  duration  than  the  short-circuit  transient  of  its  field,  the 
more  so,  the  greater  m,  that  is,  the  larger  the  ratio  of  momentary 
to  permanent  short-circuit  current. 

In  Fig.  21  the  decrease  of  the  transient  is  shown  greatly  exagger- 
ated compared  with  the  frequency  of  the  armature  currents,  and 
Fig.  22  shows  the  curves  more  nearly  in  their  actual  proportions. 

The  preceding  would  represent  the  short-circuit  phenomena,  if 
there  were  no  armature  transient.  However,  the  armature  cir- 
cuit contains  inductance  also,  that  is,  stores  magnetic  energy,  and 

thereby  gives  rise  to  a  transient,  of  duration  T  =  — ,  where  L  = 

inductance,  r  =  resistance  of  armature  circuit.  The  armature 
transient  usually  is  very  much  shorter  in  duration  than  the  field 
transient. 

The  armature  currents  thus  do  not  instantly  assume  their 
symmetrical  alternating  values,  but  if  in  Fig.  215,  iV,  iz,  is  are 
the  instantaneous  values  of  the  armature  currents  in  the  moment 
of  start,  t  —  0,  three  transients  are  superposed  upon  these,  and 


42         ELECTRIC  DISCHARGES,   WAVES  AND  IMPULSES. 

start  with  the  values  —ii,  —iz,  —is'.  The  resultant  armature 
currents  are  derived  by  the  addition  of  these  armature  transients 
upon  the  permanent  armature  currents,  in  the  manner  as  dis- 
cussed in  paragraph  18,  except  that  in  the  present  case  even  the 
permanent  armature  currents  ii,  i2,  is  are  slow  transients. 

In  Fig.  22B  are  shown  the  three  armature  short-circuit  currents, 
in  their  actual  shape  as  resultant  from  the  armature  transient 
and  the  field  transient.  The  field  transient  (or  rather  its  begin- 
ning) is  shown  as  Fig,  22 A.  Fig.  22B  gives  the  three  armature 


Fig.  22.  —  Momentary  Short  Circuit  Characteristic  of  Three-phase 
Alternator. 


currents  for  the  case  where  the  circuit  is  closed  at  the  moment  when 
t'i  should  be  maximum ;  ii  then  shows  the  maximum  transient,  and 
iz  and  ^3  transients  in  opposite  direction,  of  half  amplitude.  These 
armature  transients  rapidly  disappear,  and  the  three  currents 
become  symmetrical,  and  gradually  decrease  with  the  field  tran- 
sient to  the  final  value  indicated  in  the  figure. 

The  resultant  m.m.f.  of  three  three-phase  currents,  or  the  arma- 
ture reaction,  is  constant  if  the  currents  are  constant,  and  as  the 
currents  decrease  with  the  field  transient,  the  resultant  armature 
reaction  decreases  in  the  same  proportion  as  the  field,  as  is  shown 


SINGLE-ENERGY   TRANSIENTS.  43 

in  Fig.  21(7  by  F.  During  the  initial  part  of  the  short  circuit, 
however,  while  the  armature  transient  is  appreciable  and  the 
armature  currents  thus  unsymmetrical,  as  seen  in  Fig.  225,  their 
resultant  polyphase  m.m.f.  also  shows  a  transient,  the  transient 
of  the  rotating  magnetic  field  discussed  in  paragraph  18.  That  is, 
it  approaches  the  curve  F  of  Fig.  21  C  by  a  series  of  oscillations, 
as  indicated  in  Fig.  21E. 

Since  the  resultant  m.m.f.  of  the  machine,  which  produces  the 
flux,  is  the  difference  of  the  field  excitation,  Fig.  21  D  and  the 
armature  reaction,  then  if  the  armature  reaction  shows  an  initial  os- 
cillation, in  Fig.  21  E,  the  field-exciting  current  must  give  the  same 
oscillation,  since  its  m.m.f.  minus  the  armature  reaction  gives  the 
resultant  field  excitation  corresponding  to  flux  $>.  The  starting 
transient  of  the  polyphase  armature  reaction  thus  appears  in  the 
field  current,  as  shown  in  Fig.  22(7,  as  an  oscillation  of  full  machine 
frequency.  As  the  mutual  induction  between  armature  and  field 
circuit  is  not  perfect,  the  transient  pulsation  of  armature  reaction 
appears  with  reduced  amplitude  in  the  field  current,  and  this 
reduction  is  the  greater,  the  poorer  the  mutual  inductance,  that 
is,  the  more  distant  the  field  winding  is  from  the  armature  wind- 
ing. In  Fig.  22(7  a  damping  of  20  per  cent  is  assumed,  which 
corresponds  to  fairly  good  mutual  inductance  between  field  and 
armature,  as  met  in  turboalternators. 

If  the  field-exciting  circuit  contains  inductance  outside  of  the 
alternator  field,  as  is  always  the  case  to  a  slight  extent,  the  pul- 
sations of  the  field  current,  Fig.  22(7,  are  slightly  reduced  and 
delayed  in  phase;  and  with  considerable  inductance  intentionally 
inserted  into  the  field  circuit,  the  effect  of  this  inductance  would 
require  consideration. 

From  the  constants  of  the  alternator,  the  momentary  short- 
circuit  characteristics  can  now  be  constructed. 

Assuming  that  the  duration  of  the  field  transient  is 


sec., 


(m  —I)r0 
the  duration  of  the  armature  transient  is 

T  =  ~  =  .1  sec. 
And  assuming  that  the  armature  reaction  is  5  times  the  armature 


44         ELECTRIC  DISCHARGES,   WAVES  AND  IMPULSES. 
self-induction,  that  is,  the  synchronous  reactance  is  6  times  the  self- 
inductive  reactance,  —  =  m  =  6.     The  frequency  is  25  cycles. 

Xi 

If  <f>i  is  the  initial  or  open-circuit  flux  of  the  machine,  the  short- 

3>i      1 

circuit  flux  is  <J>0  =  —  =  ~  $1,  and  the  field  transient  $  is  a  tran- 
m      o 

sient  of  duration  1  sec.,  connecting  $1  and  <£0,  Fig.  22  A,  repre- 
sented by  the  expression 


The  permanent  armature  currents  ii,  i%,  is  then  are  currents 
starting  with  the  values  m  —  ,  and  decreasing  to  the  final  short- 

XQ 

circuit  current  —  ,  on  the  field  transient  of  duration  T0.     To  these 

XQ 

currents  are  added  the  armature  transients,  of  duration  T,  which 
start  with  initial  values  equal  but  opposite  in  sign  to  the  initial 
values  of  the  permanent  (or  rather  slowly  transient)  armature 
currents,  as  discussed  in  paragraph  18,  and  thereby  give  the  asym- 
metrical resultant  currents,  Fig.  225. 

The  field  current  i  gives  the  same  slow  transient  as  the  flux  <£, 
starting  with  i0f  =  miQ,  and  tapering  to  the  final  value  i0.  Upon 
this  is  superimposed  the  initial  full-frequency  pulsation  of  the 
armature  reaction.  The  transient  of  the  rotating  field,  of  duration 
T  =  .1  sec.,  is  constructed  as  in  paragraph  18,  and  for  its  instan- 
taneous values  the  percentage  deviation  of  the  resultant  field 
from  its  permanent  value  .is  calculated.  Assuming  20  per  cent 
damping  in  the  reaction  on  the  field  excitation,  the  instantaneous 
values  of  the  slow  field  transient  (that  is,  of  the  current  (i  —  i'0), 
since  i0  is  the  permanent  component)  then  are  increased  or  de- 
creased by  80  per  cent  of  the  percentage  variation  of  the  transient 
field  of  armature  reaction  from  uniformity,  and  thereby  the  field 
curve,  Fig.  22C,  is  derived.  Here  the  correction  for  the  external 
field  inductance  is  to  be  applied,  if  considerable. 

Since  the  transient  of  the  armature  reaction  does  not  depend 
on  the  point  of  the  wave  where  the  short  circuit  occurs,  it  follows 
that  the  phenomena  at  the  short  circuit  of  a  polyphase  alternator 
are  always  the  same,  that  is,  independent  of  the  point  of  the  wave 
at  which  the  short  circuit  occurs,  with  the  exception  of  the  initial 
wave  shape  of  the  armature  currents,  which  individually  depend 


SINGLE-ENERGY   TRANSIENTS. 


45 


on  the  point  of  the  wave  at  which  the  phenomenon  begins,  but  not 
so  in  their  resultant  effect. 

21.  The  conditions  with  a  single-phase  short  circuit  are  differ- 
ent, since  the  single-phase  armature  reaction  is  pulsating,  vary- 
ing between  zero  and  double  its  average  value,  with  double  the 
machine  frequency. 

The  slow  field  transient  and  its  effects  are  the  same  as  shown  in 
Fig.  21,  A  to  D. 

However,  the  pulsating  armature  reaction  produces  a  corre- 
sponding pulsation  in  the  field  circuit.  This  pulsation  is  of  double 


Fig.  23.  —  Symmetrical  Momentary  Single-phase  Short  Circuit  of  Alternator. 

frequency,  and  is  not  transient,  but  equally  exists  in  the  final  short- 
circuit  current. 

Furthermore,  the  armature  transient  is  not  constant  in  its 
reaction  on  the  field,  but  varies  with  the  point  of  the  wave  at 
which  the  short  circuit  starts. 

Assume  that  the  short  circuit  starts  at  that  point  of  the 
wave  where  the  permanent  (or  rather  slowly  transient)  armature 
current  should  be  zero:  then  no  armature  transient  exists,  and 
the  armature  current  is  symmetrical  from  the  beginning,  and 
shows  the  slow  transient  of  the  field,  as  shown  in  Fig.  23,  where  A 


46         ELECTRIC  DISCHARGES,   WAVES  AND  IMPULSES. 

is  the  field  transient  <i>  (the  same  as  in  Fig.  22  A)  and  B  the  arma- 
ture current,  decreasing  from  an  initial  value,  which  is  m  times 
the  final  value,  on  the  field  transient. 

Assume  then  that  the  mutual  induction  between  field  and 
armature  is  such  that  60  per  cent  of  the  pulsation  of  armature 
reaction  appears  in  the  field  current.  Forty  per  cent  damping  for 
the  double-frequency  reaction  would  about  correspond  to  the  20 
per  cent  damping  assumed  for  the  transient  full-frequency  pulsa- 
tion of  the  polyphase  machine.  The  transient  field  current  thus 
pulsates  by  60  per  cent  around  the  slow  field  transient,  as  shown 
by  Fig.  23C;  passing  a  maximum  for  every  maximum  of  armature 


Fig.  24.  —  Asymmetrical  Momentary  Single-phase  Short  Circuit  of  Alternator. 

current,  and  thus  maximum  of  armature  reaction,  and  a  minimum 
for  every  zero  value  of  armature  current,  and  thus  armature  reac- 
tion. 

Such  single-phase  short-circuit  transients  have  occasionally  been 
recorded  by  the  oscillograph,  as  shown  in  Fig.  27.  Usually,  how- 
ever, the  circuit  is  closed  at  a  point  of  the  wave  where  the  perma- 
nent armature  current  would  not  be  zero,  and  an  armature  transient 
appears,  with  an  initial  value  equal,  but  opposite  to,  the  initial 
value  of  the  permanent  armature  current.  This  is  shown  in 
Fig.  24  for  the  case  of  closing  the  circuit  at  the  moment  where  the 


SINGLE-ENERGY  TRANSIENTS.  47 

armature  current  should  be  a  maximum,  and  its  transient  thus  a 
maximum.  The  field  transient  <£  is  the  same  as  before.  The 
armature  current  shows  the  initial  asymmetry  resulting  from  the 
armature  transient,  and  superimposed  on  the  slow  field  transient. 

On  the  field  current,  which,  due  to  the  single-phase  armature 
reaction,  shows  a  permanent  double-frequency  pulsation,  is  now 
superimposed  the  transient  full-frequency  pulsation  resultant  from 
the  transient  armature  reaction,  as  discussed  in  paragraph  20. 
Every  second  peak  of  the  permanent  double-frequency  pulsation 
then  coincides  with  a  peak  of  the  transient  full-frequency  pulsa- 
tion, and  is  thereby  increased,  while  the  intermediate  peak  of  the 
double-frequency  pulsation  coincides  with  a  minimum  of  the  full- 
frequency  pulsation,  and  is  thereby  reduced.  The  result  is  that 
successive  waves  of  the  double-frequency  pulsation  of  the  field 
current  are  unequal  in  amplitude,  and  high  and  low  peaks  alter- 
nate. The  difference  between  successive  double-frequency  waves 
is  a  maximum  in  the  beginning,  and  gradually  decreases,  due  to 
the  decrease  of  the  transient  full-frequency  pulsation,  and  finally 
the  double-frequency  pulsation  becomes  symmetrical,  as  shown  in 
Fig.  24C. 

In  the  particular  instance  of  Fig.  24,  the  double-frequency  and 
the  full-frequency  peaks  coincide,  and  the  minima  of  the  field- 
current  curve  thus  are  symmetrical.  If  the  circuit  were  closed  at 
another  point  of  the  wave,  the  double-frequency  minima  would 
become  unequal,  and  the  maxima  more  nearly  equal,  as  is  easily 
seen. 

While  the  field-exciting  current  is  pulsating  in  a  manner  deter- 
mined by  the  full-frequency  transient  and  double-frequency  per- 
manent armature  reaction,  the  potential  difference  across  the 
field  winding  may  pulsate  less,  if  little  or  no  external  resistance 
or  inductance  is  present,  or  may  pulsate  so  as  to  be  nearly  alter- 
nating and  many  times  higher  than  the  exciter  voltage,  if  consid- 
erable external  resistance  or  inductance  is  present;  and  therefore 
it  is  not  characteristic  of  the  phenomenon,  but  may  become  impor- 
tant by  its  disruptive  effects,  if  reaching  very  high  values  of  voltage. 

With  a  single-phase  short  circuit  on  a  polyphase  machine,  the 
double-frequency  pulsation  of  the  field  resulting  from  the  single- 
phase  armature  reaction  induces  in  the  machine  phase,  which  is 
in  quadrature  to  the  short-circuited  phase,  an  e.m.f.  which  con- 
tains the  frequencies  /(2  ±  1),  that  is,  full  frequency  and  triple 


48        ELECTRIC  DISCHARGES,   WAVES  AND  IMPULSES. 


SINGLE-ENERGY   TRANSIENTS.  49 

frequency,  and  as  the  result  an  increase  of  voltage  and  a  distor- 
tion of  the  quadrature  phase  occurs,  as  shown  in  the  oscillogram 
Fig.  25. 

Various  momentary  short-circuit  phenomena  are  illustrated  by 
the  oscillograms  Figs.  26  to  28. 

Figs.  26A  and  265  show  the  momentary  three-phase  short  cir- 
cuit of  a  4-polar  25-cycle  1500-kw.  steam  turbine  alternator.  The 


Fig.  26 A.  —  CD9399.  —  Symmetrical. 


Fig.  2QB.  —  CD9397.  —  Asymmetrical.  Momentary  Three-phase  Short  Cir- 
cuit of  1500-Kw.  2300- Volt  Three-phase  Alternator  (ATB-4-1500-1800) . 
Oscillograms  of  Armature  Current  and  Field  Current. 

lower  curve  gives  the  transient  of  the  field-exciting  current,  the 
upper  curve  that  of  one  of  the  armature  currents,  —  in  Fig.  26A 
that  current  which  should  be  near  zero,  in  Fig.  26B  that  which 
should  be  near  its  maximum  value  at  the  moment  where  the  short 
circuit  starts. 

Fig.  27  shows  the  single-phase  short  circuit  of  a  pair  of  machines 
in  which  the  short  circuit  occurred  at  the  moment  in  which  the 
armature  short-circuit  current  should  be  zero;  the  armature  cur- 


50         ELECTRIC  DISCHARGES,   WAVES  AND  IMPULSES. 

rent  wave,  therefore,  is  symmetrical,  and  the  field  current  shows 
only  the  double-frequency  pulsation.  Only  a  few  half-waves  were 
recorded  before  the  circuit  breaker  opened  the  short  circuit. 


Fig.  27.  —  CD5128.  —  Symmetrical.  Momentary  Single-phase  Short  Circuit 
of  Alternator.  Oscillogram  of  Armature  Current,  Armature  Voltage, 
and  Field  Current. 


Fig.  28.  —  CD6565.  — Asymmetrical.  Momentary  Single-phase  Short  Circuit 
of  5000-Kw.  11, 000- Volt  Three-phase  Alternator  (ATB-6-5000-500) . 
Oscillogram  of  Armature  Current  and  Field  Current. 

Fig.  28  shows  the  single-phase  short  circuit  of  a  6-polar  5000-kw. 
11,000-volt  steam  turbine  alternator,  which  occurred  at  a  point  of 
the  wave  where  the  armature  current  should  be  not  far  from  its 
maximum.  The  transient  armature  current,  therefore,  starts  un- 


SINGLE-ENERGY  TRANSIENTS.  51 

symmetrical,  and  the  double-frequency  pulsation  of  the  field  cur- 
rent shows  during  the  first  few  cycles  the  alternate  high  and  low 
peaks  resulting  from  the  superposition  of  the  full-frequency  tran- 
sient pulsation  of  the  rotating  magnetic  field  of  armature  reaction. 
Interesting  in  this  oscillogram  is  the  irregular  initial  decrease  of  the 
armature  current  and  the  sudden  change  of  its  wave  shape,  which 
is  the  result  of  the  transient  of  the  current  transformer,  through 
which  the  armature  current  was  recorded.  On  the  true  armature- 
current  transient  superposes  the  starting  transient  of  the  current 
transformer. 

Fig.  25  shows  a  single-phase  short  circuit  of  a  quarter-phase 
alternator;  the  upper  wave  is  the  voltage  of  the  phase  which  is 
not  short-circuited,  and  shows  the  increase  and  distortion  resulting 
from  the  double-frequency  pulsation  of  the  armature  reaction. 


LECTURE  V. 

SINGLE-ENERGY  TRANSIENT  OF  IRONCLAD 
CIRCUIT. 

22.  Usually  in  electric  circuits,  current,  voltage,  the  magnetic 
field  and  the  dielectric  field  are  proportional  to  each  other,  and  the 
transient  thus  is  a  simple  exponential,  if  resulting  from  one  form  of 
stored  energy,  as  discussed  in  the  preceding  lectures.  This,  how- 
ever, is  no  longer  the  case  if  the  magnetic  field  contains  iron  or 
other  magnetic  materials,  or  if  the  dielectric  field  reaches  densities 
beyond  the  dielectric  strength  of  the  carrier  of  the  field,  etc. ;  and 
the  proportionality  between  current  or  voltage  and  their  respective 
fields,  the  magnetic  and  the  dielectric,  thus  ceases,  or,  as  it  may  be 
expressed,  the  inductance  L  is  not  constant,  but  varies  with  the 
current,  or  the  capacity  is  not  constant,  but  varies  with  the  voltage. 

The  most  important  case  is  that  of  the  ironclad  magnetic  cir- 
cuit, as  it  exists  in  one  of  the  most  important  electrical  apparatus, 
the  alternating-current  transformer.  If  the  iron  magnetic  circuit 
contains  an  air  gap  of  sufficient  length,  the  magnetizing  force  con- 
sumed in  the  iron,  below  magnetic  saturation,  is  small  compared 
with  that  consumed  in  the  air  gap,  and  the  magnetic  flux,  therefore, 
is  proportional  to  the  current  up  to  the  values  where  magnetic 
saturation  begins.  Below  saturation  values  of  current,  the  tran- 
sient thus  is  the  simple  exponential  discussed  before. 

If  the  magnetic  circuit  is  closed  entirely  by  iron,  the  magnetic 
flux  is  not  proportional  to  the  current,  and  the  inductance  thus  not 
constant,  but  varies  over  the  entire  range  of  currents,  following 
the  permeability  curve  of  the  iron.  Furthermore,  the  transient 
due  to  a  decrease  of  the  stored  magnetic  energy  differs  in  shape 
and  in  value  from  that  due  to  an  increase  of  magnetic  energy,  since 
the  rising  and  decreasing  magnetization  curves  differ,  as  shown  by 
the  hysteresis  cycle. 

Since  no  satisfactory  mathematical  expression  has  yet  been 
found  for  the  cyclic  curve  of  hysteresis,  a  mathematical  calcula- 
tion is  not  feasible,, but  the  transient  has  to  be  calculated  by  an 
'^''"r  '*_/ ? :,":  \  :  52 


SINGLE-ENERGY  TRANSIENT  OF  IRONCLAD  CIRCUIT.      53 

approximate  step-by-step  method,  as  illustrated  for  the  starting 
transient  of  an  alternating-current  transformer  in  "Transient  Elec- 
tric Phenomena  and  Oscillations,"  Section  I,  Chapter  XII.  Such 
methods  are  very  cumbersome  and  applicable  only  to  numerical 
instances. 

An  approximate  calculation,  giving  an  idea  of  the  shape  of  the 
transient  of  the  ironclad  magnetic  circuit,  can  be  made  by  neglect- 
ing the  difference  between  the  rising  and  decreasing  magnetic 
characteristic,  and  using  the  approximation  of  the  magnetic  char- 
acteristic given  by  Frohlich's  formula: 


which  is  usually  represented  in  the  form  given  by  Kennelly: 

T/> 

p  =  -  =  a  +  crOC;  (2) 

that  is,  the  reluctivity  is  a  linear  function  of  the  field  intensity. 
It  gives  a  fair  approximation  for  higher  magnetic  densities. 

This  formula  is  based  on  the  fairly  rational  assumption  that  the 
permeability  of  the  iron  is  proportional  to  its  remaining  magnetiza- 
bility.  That  is,  the  magnetic-flux  density  (B  consists  of  a  compo- 
nent 3C,  the  field  intensity,  which  is  the  flux  density  in  space,  and 
a  component  (B'  =  (B  —  3C,  which  is  the  additional  flux  density 
carried  by  the  iron.  (B'  is  frequently  called  the  "  metallic-flux 
density."  With  increasing  3C,  (B'  reaches  a  finite  limiting  value, 
which  in  iron  is  about 

&x'  =  20,000  lines  per  cm2.  * 

At  any  density  (B',  the  remaining  magnetizability  then  is 
(B^'  —  (B',  and,  assuming  the  (metallic)  permeability  as  proportional 
hereto,  gives 

and,  substituting 


gives 

a,  =  cftco'rc^ 

*  See  "On  the  Law  of  Hysteresis,"  Part  II,  A.I.E.E.  Transactions,  1892, 
page  621. 


54        ELECTRIC  DISCHARGES,   WAVES  AND  IMPULSES. 
or,  substituting 


1_         1 

***        /  t*«      ,—fc        /  (/  • 


gives  equation  (1). 

For  OC  =  0  in  equation  (1),  ^  =  -  ;  for  3C  =  oo    »  =  -  ;  that  is, 

uv       a:  cr 

in  equation  (1),  -  =  initial  permeability,  -  =  saturation  value  of 

Oi  (7 

magnetic  density. 

If  the  magnetic  circuit  contains  an  air  gap,  the  reluctance  of  the 
iron  part  is  given  by  equation  (2),  that  of  the  air  part  is  constant, 
and  the  total  reluctance  thus  is 

p  =  ft  +  ffK, 

where  3  =  a  plus  the  reluctance  of  the  air  gap.  Equation  (1), 
therefore,  remains  applicable,  except  that  the  value  of  a  is  in- 
creased. 

In  addition  to  the  metallic  flux  given  by  equation  (1),  a  greater 
or  smaller  part  of  the  flux  always  passes  through  the  air  or  through 
space  in  general,  and  then  has  constant  permeance,  that  is,  is  given 
by 


23.  In  general,  the  flux  in  an  ironclad  magnetic  circuit  can, 
therefore,  be  represented  as  function  of  the  current  by  an  expression 
of  the  form 


where         ,  .  =  &  is  that  part  of  the  flux  which  passes  through 
1  -f-  ut 

the  iron  and  whatever  air  space  may  be  in  series  with  the  iron, 
and  a  is  the  part  of  the  flux  passing  through  nonmagnetic 
material. 

Denoting  now 


L2  =  nc  10-8,  i 

where  n  =  number  of  turns  of  the  electric  circuit,  which  is  inter- 
linked with  the  magnetic  circuit,  L2  is  the  inductance  of  the  air 
part  of  the  magnetic  circuit,  LI  the  (virtual)  initial  inductance,  that 
is,  inductance  at  very  small  currents,  of  the  iron  part  of  the  mag- 


SINGLE-ENERGY  TRANSIENT  OF  IRONCLAD  CIRCUIT.      55 
netic  circuit,  and  =-  the  saturation  value  of  the  flux  in  the  iron. 

72,CJ>'  d 

That  is,  for  i  =  0,  —  r-  =  Z/i  ;  and  for  i  =  oo  ,  <£'  =  T  . 

i  0 

If  r  =  resistance,  the  duration  of  the  component  of  the  transient 
resulting  from  the  air  flux  would  be 

_  L2      nc  10~8 

*V-7"    T~ 

and  the  duration  of  the  transient  which  would  result  from  the 
initial  inductance  of  the  iron  flux  would  be 


The  differential  equation  of  the  transient  is:  induced  voltage 
plus  resistance  drop  equal  zero  ;  that  is, 


Substituting  (3)  and  differentiating  gives 
na  10~8   di  .        .,_   a  di  '  . 

(i+Wdi  +  ncl0rSdt+ 

and,  substituting  (5)  and  (6), 


t(l  +  bi)2     Z5  d*  ' 

hence,  separating  the  variables, 

Tidi      +  Tidi  +  dt  =  Q 

The  first  term  is  integrated  by  resolving  into  partial  fractions 

1  1          6  6 

i(l  +  6i)2  "  i      1  +  6i      (1  +  6i)2>. 

and  the  integration  of  differential  equation  (7)  then  gives 


If  then,  for  the  time  t  =  tQ,  the  current  is  i  =  i0,  these  values 
substituted  in  (8)  give  the  integration  constant  C: 

T1log-        +  !T2logio  +  T-  +  ^o  +  C  =  0,          (9) 


56         ELECTRIC  DISCHARGES,   WAVES  AND  IMPULSES. 
and,  subtracting  (8)  from  (9),  gives 


1  +  6i  5 ' 


(10) 


This  equation  is  so  complex  in  i  that  it  is  not  possible  to  cal- 
culate from  the  different  values  of  t  the  corresponding  values  of  i; 
but  inversely,  for  different  values  of  i  the  corresponding  values 
of  t  can  be  calculated,  and  the  corresponding  values  of  i  and  t, 
derived  in  this  manner,  can  be  plotted  as  a  curve,  which  gives  the 
single-energy  transient  of  the  ironclad  magnetic  circuit. 


Tra 


sient  o 


Ironclad  Inductive  Circuit : 


t=2.92- 


i  + 


t-.6i  j 


l+.6i 


(dotted:  t  =  1.0851g  i— .50?) 


2  3  4  5 

Fig.  29. 


6     seconds 


Such  is  done  in  Fig.  29,  for  the  values  of  the  constants 


a  =  4  X  105, 
c  =  4  X  104, 
b  =  .6, 
n  =  300. 


SINGLE-ENERGY  TRANSIENT  OF  IRONCLAD  CIRCUIT.      57 


58        ELECTRIC  DISCHARGES,   WAVES  AND  IMPULSES. 
This  gives 

T  =  4 

Assuming  i0  =  10  amperes  for  t0  =  0,  gives  from  (10)  the  equa- 
tion : 

4 


T  =  2.92  -  1  9.21  log10       ^  +  . 


921 


.6  i 


Herein,  the  logarithms  have  been  reduced  to  the  base  10  by 
division  with  logwe  =  .4343. 

For  comparison  is  shown,  in  dotted  line,  in  Fig.  29,  the  transient 
of  a  circuit  containing  no  iron,  and  of  such  constants  as  to  give 
about  the  same  duration: 

t  =  1.0S5log™i-  .507. 

As  seen,  in  the  ironclad  transient  the  current  curve  is  very 
much  steeper  in  the  range  of  high  currents,  where  magnetic  sat- 
uration is  reached,  but  the  current  is  slower  in  the  range  of  medium 
magnetic  densities. 

Thus,  in  ironclad  transients  very  high-current  values  of  short 
duration  may  occur,  and  such  transients,  as  those  of  the  starting 
current  of  alternating-current  transformers,  may  therefore  be  of 
serious  importance  by  their  excessive  current  values. 

An  oscillogram  of  the  voltage  and  current  waves  in  an  11,000-kw. 
high-voltage  60-cycle  three-phase  transformer,  when  switching  onto 
the  generating  station  near  the  most  unfavorable  point  of  the 
wave,  is  reproduced  in  Fig.  30.  As  seen,  an  excessive  current  rush 
persists  for  a  number  of  cycles,  causing  a  distortion  of  the  volt- 
age wave,  and  the  current  waves  remain  unsymmetrical  for  many 
cycles. 


LECTURE  VI. 
DOUBLE-ENERGY   TRANSIENTS. 

24.  In  a  circuit  in  which  energy  can  be  stored  in  one  form  only, 
the  change  in  the  stored  energy  which  can  take  place  as  the  result 
of  a  change  of  the  circuit  conditions  is  an  increase  or  decrease. 
The  transient  can  be  separated  from  the  permanent  condition,  and 
then  always  is  the  representation  of  a  gradual  decrease  of  energy. 
Even  if  the  stored  energy  after  the  change  of  circuit  conditions  is 
greater  than  before,  and  during  the  transition  period  an  increase 
of  energy  occurs,  the  representation  still  is  by  a  decrease  of  the 
transient.  This  transient  then  is  the  difference  between  the  energy 
storage  in  the  permanent  condition  and  the  energy  storage  during 
the  transition  period. 

If  the  law  of  proportionality  between  current,  voltage,  magnetic 
flux,  etc.,  applies,  the  single-energy  transient  is  a  simple  exponential 

function : 

j_ 

y  =  i/oe    T°,  (1) 

where 

?/o  =  initial  value  of  the  transient,  and 
TO  =  duration  of  the  transient, 

that  is,  the  time  which  the  transient  voltage,  current,  etc.,  would 
last  if  maintained  at  its  initial  value. 

The  duration  T0  is  the  ratio  of  the  energy-storage  coefficient 
to  the  power-dissipation  coefficient.  Thus,  if  energy  is  stored  by 
the  current  i,  as  magnetic  field, 

T0  =  £,  (2) 

where  L  =  inductance  =  coefficient  of  energy  storage  by  the  cur- 
rent, r  =  resistance  =  coefficient  of  power  dissipation  by  the  current. 
If  the  energy  is  stored  by  the  voltage  e,  as  dielectric  field,  the 
duration  of  the  transient  would  be 

TJ  =  -,  (3) 

s/ 
59 


60        ELECTRIC  DISCHARGES,   WAVES  AND  IMPULSES. 

where  C  =  capacity  =  coefficient  of  energy  storage  by  the  volt- 
age, in  the  dielectric  field,  and  g  =  conductance  =  coefficient  of 
power  consumption  by  the  voltage,  as  leakage  conductance  by 
the  voltage,  corona,  dielectric  hysteresis,  etc. 

Thus  the  transient  of  the  spontaneous  discharge  of  a  condenser 
would  be  represented  by 

e  =  e0e~£ct.  (4) 

Similar  single-energy  transients  may  occur  in  other  systems. 
For  instance,  the  transient  by  which  a  water  jet  approaches  con- 
stant velocity  when  falling  under  gravitation  through  a  resisting 
medium  would  have  the  duration 

T  =  -°,  (5) 

where  VQ  =  limiting  velocity,  g  =  acceleration  of  gravity,  and  would 
be  given  by 

v  =  v0(l-6~r}.  (6) 

In  a  system  in  which  energy  can  be  stored  in  two  different 
forms,  as  for  instance  as  magnetic  and  as  dielectric  energy  in  a 
circuit  containing  inductance  and  capacity,  in  addition  to  the 
gradual  decrease  of  stored  energy  similar  to  that  represented  by 
the  single-energy  transient,  a  transfer  of  energy  can  occur  between 
its  two  different  forms. 

Thus,  if  i  =  transient  current,  e  =  transient  voltage  (that  is, 
the  difference  between  the  respective  currents  and  voltages  exist- 
ing in  the  circuit  as  result  of  the  previous  circuit  condition,  and 
the  values  which  should  exist  as  result  of  the  change  of  circuit 
conditions),  then  the  total  stored  energy  is 

w      Li*      Ce*    ) 

'T+-2-'£  (7) 

=    Wm  +Wd.     > 

While  the  total  energy  W  decreases  by  dissipation,  Wm  may  be 
converted  into  Wd,  or  inversely. 

Such  an  energy  transfer  may  be  periodic,  that  is,  magnetic  energy 
may  change  to  dielectric  and  then  back  again;  or  unidirectional, 
that  is,  magnetic  energy  may  change  to  dielectric  (or  inversely, 
dielectric  to  magnetic),  but  never  change  back  again;  but  the 


DOUBLE-ENERGY  TRANSIENTS.  61 

energy  is  dissipated  before  this.  This  latter  case  occurs  when  the 
dissipation  of  energy  is  very  rapid,  the  resistance  (or  conductance) 
high,  and  therefore  gives  transients,  which  rarely  are  of  industrial 
importance,  as  they  are  of  short  duration  and  of  low  power.  It 
therefore  is  sufficient  to  consider  the  oscillating  double-energy 
transient,  that  is,  the  case  in  which  the  energy  changes  periodically 
between  its  two  forms,  during  its  gradual  dissipation. 

This  may  be  done  by  considering  separately  the  periodic  trans- 
fer, or  pulsation  of  the  energy  between  its  two  forms,  and  the 
gradual  dissipation  of  energy. 

A  .   Pulsation  of  energy. 

25.  The  magnetic  energy  is  a  maximum  at  the  moment  when 
the  dielectric  energy  is  zero,  and  when  all  the  energy,  therefore,  is 
magnetic  ;  and  the  magnetic  energy  is  then 


where  t'0  =  maximum  transient  current. 

The  dielectric  energy  is  a  maximum  at  the  moment  when  the 
magnetic  energy  is  zero,  and  all  the  energy  therefore  dielectric, 
and  is  then 

Ce02 

2  ' 

where  e0  =  maximum  transient  voltage. 

As  it  is  the  same  stored  energy  which  alternately  appears  as 
magnetic  and  as  dielectric  energy,  it  obviously  is 

W  _  Ceo2 
~2~      ~2" 

This  gives  a  relation  between  the  maximum  transient  current 
and  the  maximum  transient  voltage: 


v/: 


-^  therefore  is  of  the  nature  of  an  impedance  z0,  and  is  called 
the  natural  impedance,  or  the  surge  impedance,  of  the  circuit ;  and 

fc 

its  reciprocal,  V/y  =  yo,  is  the  natural  admittance,  or  the  surge 

T      J  j 

admittance,  of  the  circuit. 


62        ELECTRIC  DISCHARGES,   WAVES  AND  IMPULSES. 

The  maximum  transient  voltage  can  thus  be  calculated  from 
the  maximum  transient  current: 

#0   =  'Z'O  V/  7>  =  i&Qj  (10) 

and  inversely, 

/C 
io  =  eo  y  j  =  e02/o.  (11) 

This  relation  is  very  important,  as  frequently  in  double-energy 
transients  one  of  the  quantities  e$  or  i0  is  given,  and  it  is  impor- 
tant to  determine  the  other. 

For  instance,  if  a  line  is  short-circuited,  and  the  short-circuit 
current  IQ  suddenly  broken,  the  maximum  voltage  which  can  be 
induced  by  the  dissipation  of  the  stored  magnetic  energy  of  the 
short-circuit  current  is  e0  =  igZo. 

If  one  conductor  of  an  ungrounded  cable  system  is  grounded, 
the  maximum  momentary  current  which  may  flow  to  ground  is 
io  =  eo2/o,  where  e0  =  voltage  between  cable  conductor  and  ground. 

If  lightning  strikes  a  line,  and  the  maximum  voltage  which  it 
may  produce  on  the  line,  as  limited  by  the  disruptive  strength  of 
the  line  insulation  against  momentary  voltages,  is  e0,  the  maximum 
discharge  current  in  the  line  is  limited  to  i0  =  e<>yo. 

If  L  is  high  but  C  low,  as  in  the  high-potential  winding  of  a 
high-voltage  transformer  (which  winding  can  be  considered  as  a 
circuit  of  distributed  capacity,  inductance,  and  resistance),  z0  is 
high  and  T/O  low.  That  is,  a  high  transient  voltage  can  produce 
only  moderate  transient  currents,  but  even  a  small  transient  cur- 
rent produces  high  voltages.  Thus  reactances,  and  other  reactive 
apparatus,  as  transformers,  stop  the  passage  of  large  oscillating 
currents,  but  do  so  by  the  production  of  high  oscillating  voltages. 

Inversely,  if  L  is  low  and  C  high,  as  in  an  underground  cable, 
ZQ  is  low  but  2/0  high,  and  even  moderate  oscillating  voltages  pro- 
duce large  oscillating  currents,  but  even  large  oscillating  currents 
produce  only  moderate  voltages.  Thus  underground  cables  are 
little  liable  to  the  production  of  high  oscillating  voltages.  This 
is  fortunate,  as  the  dielectric  strength  of  a  cable  is  necessarily 
relatively  much  lower  than  that  of  a  transmission  line,  due  to 
the  close  proximity  of  the  conductors  in  the  former.  A  cable, 
therefore,  when  receiving  the  moderate  or  small  oscillating  cur- 
rents which  may  originate  in  a  transformer,  gives  only  very  low 


DOUBLE-ENERGY   TRANSIENTS.  63 

oscillating  voltages,  that  is,  acts  as  a  short  circuit  for  the  trans- 
former oscillation,  and  therefore  protects  the  latter.  Inversely, 
if  the  large  oscillating  current  of  a  cable  enters  a  reactive  device, 
as  a  current  transformer,  it  produces  enormous  voltages  therein. 
Thus,  cable  oscillations  are  more  liable  to  be  destructive  to  the 
reactive  apparatus,  transformers,  etc.,  connected  with  the  cable, 
than  to  the  cable  itself. 

A  transmission  line  is  intermediate  in  the  values  of  z0  and  yQ 
between  the  cable  and  the  reactive  apparatus,  thus  acting  like  a 
reactive  apparatus  to  the  former,  like  a  cable  toward  the  latter. 
Thus,  the  transformer  is  protected  by  the  transmission  line  in 
oscillations  originating  in  the  transformer,  but  endangered  by  the 
transmission  line  in  oscillations  originating  in  the  transmission 
line. 

The  simple  consideration  of  the  relative  values  of  ZQ  =  V  ^  in 

the  different  parts  of  an  electric  system  thus  gives  considerable 
information  on  the  relative  danger  and  protective  action  of  the 
parts  on  each  other,  and  shows  the  reason  why  some  elements,  as 
current  transformers,  are  far  more  liable  to  destruction  than  others; 
but  also  shows  that  disruptive  effects  of  transient  voltages, 
observed  in  one  apparatus,  may  not  and  very  frequently  do  not 
originate  in  the  damaged  apparatus,  but  originate  in  another 
part  of  the  system,  in  which  they  were  relatively  harmless,  and 
become  dangerous  only  when  entering  the  former  apparatus. 

26.  If  there  is  a  periodic  transfer  between  magnetic  and  dielec- 
tric energy,  the  transient  current  i  and  the  transient  voltage  e 
successively  increase,  decrease,  and  become  zero. 

The  current  thus  may  be  represented  by 

i  =  locosfa  -7),  (12) 

where  iQ  is  the  maximum  value  of  current,  discussed  above,  and 

<t>    =    27Tft,  (13) 

where  /  =  the  frequency  of  this  transfer  (which  is  still  undeter- 
mined), and  7  the  phase  angle  at  the  starting  moment  of  the 
transient;  that  is, 

ii  =  IQ  cos  7  =  initial  transient  current.  (14) 

As  the  current  i  is  a  maximum  at  the  moment  when  the  magnetic 
energy  is  a  maximum  and  the  dielectric  energy  zero,  the  voltage  e 


64        ELECTRIC  DISCHARGES,   WAVES  AND  IMPULSES. 

must  be  zero  when  the  current  is  a  maximum,  and  inversely;  and 
if  the  current  is  represented  by  the  cosine  function,  the  voltage 
thus  is  represented  by  the  sine  function,  that  is, 

e  =  e0  sin  (0  -  7),  (15) 

where 

ei  =  —  e0  sin  7  =  initial  value  of  transient  voltage.         (16) 

The  frequency  /  is  still  unknown,  but  from  the  law  of  propor- 
tionality it  follows  that  there  must  be  a  frequency,  that  is,  the  suc- 
cessive conversions  between  the  two  forms  of  energy  must  occur  in 
equal  time  intervals,  for  this  reason:  If  magnetic  energy  converts 
to  dielectric  and  back  again,  at  some  moment  the  proportion  be- 
tween the  two  forms  of  energy  must  be  the  same  again  as  at  the 
starting  moment,  but  both  reduced  in  the  same  proportion  by  the 
power  dissipation.  From  this  moment  on,  the  same  cycle  then  must 
repeat  with  proportional,  but  proportionately  lowered  values. 


Fig.  31.  —  CD10017.  —  Oscillogram  of  Stationary  Oscillation  of  Varying 
Frequency:  Compound  Circuit  of  Step-up  Transformer  and  28  Miles  of 
100,000-volt  Transmission  Line. 

If,  however,  the  law  of  proportionality  does  not  exist,  the  oscil- 
lation may  not  be  of  constant  frequency.  Thus  in  Fig.  31  is  shown 
an  oscillogram  of  the  voltage  oscillation  of  the  compound  circuit 
consisting  of  28  miles  of  100,000-volt  transmission  line  and  the 
2500-kw.  high-potential  step-up  transformer  winding,  caused  by 
switching  transformer  and  28-mile  line  by  low-tension  switches  off 
a  substation  at  the  end  of  a  153-mile  transmission  line,  at  88  kv. 
With  decreasing  voltage,  the  magnetic  density  in  the  transformer 


DOUBLE-ENERGY   TRANSIENTS.  65 

decreases,  and  as  at  lower  magnetic  densities  the  permeability  of 
the  iron  is  higher,  with  the  decrease  of  voltage  the  permeability  of 
the  iron  and  thereby  the  inductance  of  the  electric  circuit  inter- 
linked with  it  increases,  and,  resulting  from  this  increased  magnetic 
energy  storage  coefficient  L,  there  follows  a  slower  period  of  oscil- 
lation, that  is,  a  decrease  of  frequency,  as  seen  on  the  oscillogram, 
from  55  cycles  to  20  cycles  per  second. 

If  the  energy  transfer  is  not  a  simple  sine  wave,  it  can  be  repre- 
sented by  a  series  of  sine  waves,  and  in  this  case  the  above  equa- 
tions (12)  and  (15)  would  still  apply,  but  the  calculation  of  the 
frequency  /  would  give  a  number  of  values  which  represent  the 
different  component  sine  waves. 

The  dielectric  field  of  a  condenser,  or  its  "  charge,"  is  capacity 
times  voltage:  Ce.  It  is,  however,  the  product  of  the  current 
flowing  into  the  condenser,  and  the  time  during  which  this  current 
flows  into  it,  that  is,  it  equals  i  t. 

Applying  the  law 

Ce  =  it  (17) 

to  the  oscillating  energy  transfer:  the  voltage  at  the  condenser 
changes  during  a  half-cycle  from  —  eQ  to  -fe0,  and  the  condenser 
charge  thus  is 

2e0C; 

2 
the  current  has  a  maximum  value  i'0,  thus  an  average  value  -i0, 

IT 

and  as  it  flows  into  the  condenser  during  one-half  cycle  of  the 
frequency  /,  that  is,  during  the  time  =-},  it  is 

2eQC  =  -io  o7» 

7T         2J 

which  is  the  expression  of  the  condenser  equation  (17)  applied  to 
the  oscillating  energy  transfer. 
Transposed,  this  equation  gives 


and  substituting  equation  (10)  into  (18),  and  canceling  with  i0, 
gives 


66         ELECTRIC  DISCHARGES,   WAVES  AND  IMPULSES. 
as  the  expression  of  the  frequency  of  the  oscillation,  where 

a  =  VLC  (20) 

is  a  convenient  abbreviation  of  the  square  root. 

The  transfer  of  energy  between  magnetic  and  dielectric  thus 

occurs  with  a  definite  frequency  /  =  ~ — - ,  and  the  oscillation  thus 

Z  TTCT 

is  a  sine  wave  without  distortion,  as  long  as  the  law  of  proportion- 
ality applies.  When  this  fails,  the  wave  may  be  distorted,  as  seen 
on  the  oscillogram  Fig.  31. 

The  equations  of  the  periodic  part  of  the  transient  can  now  be 
written  down  by  substituting  (13),  (19),  (14),  and  (16)  into  (12) 
and  (15)  : 


i  =  io  cos  (0  —  7)  =  io  cos  7  cos  <j>  +  i0  sin  7  sin 
and  by  (11): 


t  IQ   .     t 

—  i\  cos e\  —  sin  - , 

(7  €Q          <J 


i  =  i\  cos 1/001  sin  - ,  (21) 

(T  ff 

and  in  the  same  manner: 

e  =  e\  cos  -  +  z0ii  sin  - ,  (22) 

(7  a 

where  e\  is  the  initial  value  of  transient  voltage,  ii  the  initial  value 
of  transient  current. 
B.   Power  dissipation. 

27.   In  Fig.  32  are  plotted  as  A  the  periodic  component  of  the 
oscillating  current  i,  and  as  B  the  voltage  e,  as  C  the  stored  mag- 
Li2  Ce2 
netic  energy  —^  ,  and  as  D  the  stored  dielectric  energy  —  • 
z  z 

As  seen,  the  stored  magnetic  energy  pulsates,  with  double 
frequency,  2/,  between  zero  and  a  maximum,  equal  to  the  total 
stored  energy.  The  average  value  of  the  stored  magnetic  energy 
thus  is  one-half  of  the  total  stored  energy,  and  the  dissipation  of 
magnetic  energy  thus  occurs  at  half  the  rate  at  which  it  would 
occur  if  all  the  energy  were  magnetic  energy;  that  is,  the  transient 
resulting  from  the  power  dissipation  of  the  magnetic  energy  lasts 
twice  as  long  as  it  would  if  all  the  stored  energy  were  magnetic, 
or  in  other  words,  if  the  transient  were  a  single  (magnetic)  energy 


DOUBLE-ENERGY  TRANSIENTS. 


67 


transient.     In  the  latter  case,  the  duration  of  the  transient  would 
be 


and  with  only  half  the  energy  magnetic,  the  duration  thus  is 

twice  as  long,  or 

2  T 
7\=2T0  =  ^=,  (23) 

and  hereby  the  factor 


multiplies  with  the  values  of  current  and  voltage  (21)  and  (22). 


/C 


Fig.  32.  —  Relation  of  Magnetic  and  Dielectric  Energy  of  Transient. 

The  same  applies  to  the  dielectric  energy.  If  all  the  energy 
were  dielectric,  it  would  be  dissipated  by  a  transient  of  the  dura- 
tion: 

rp  f         k  • 

IV--; 


68        ELECTRIC  DISCHARGES,   WAVES  AND  IMPULSES. 

as  only  half  the  energy  is  dielectric,  the  dissipation  is  half  as  rapid, 
that  is,  the  dielectric  transient  has  the  duration 

T2  =  2  T0'  =  —  ,  (24) 

y 
and  therefore  adds  the  factor 


to  the  equations  (21)  and  (22). 

While  these  equations  (21)  and  (22)  constitute  the  periodic 
part  of  the  phenomenon,  the  part  which  represents  the  dissipa- 
tion of  power  is  given  by  the  factor 

_JL   _JL  t(±  +  ±\ 

hk  =  €      T^      T,=  €         \T^TJt  (25) 

The  duration  of  the  double-energy  transient,  T,  thus  is  given  by 


I..!    !_ 

T      IV  2V 

1/1         I 


(26) 


and  this  is  the  harmonic  mean  of  the  duration  of  the  single-energy 
magnetic  and  the  single-energy  dielectric  transient. 
It  is,  by  substituting  for  T0  and  TV, 


where  u  is  the  abbreviation  for  the  reciprocal  of  the  duration  of 
the  double-energy  transient. 

Usually,  the  dissipation  exponent  of  the  double-energy  transient 


is  given  as 

r 
2L' 

This  is  correct  only  if  g  =  0,  that  is,  the  conductance,  which  rep- 
resents the  power  dissipation  resultant  from  the  voltage  (by  leak- 
age, dielectric  induction  and  dielectric  hysteresis,  corona,  etc.), 
is  negligible.  Such  is  the  case  in  most  power  circuits  and  trans- 
mission lines,  except  at  the  highest  voltages,  where  corona  appears. 
It  is  not  always  the  case  in  underground  cables,  high-potential 


DOUBLE-ENERGY   TRANSIENTS. 


69 


transformers,  etc.,  and  is  not  the  case  in  telegraph  or  telephone 
lines,  etc.  It  is  very  nearly  the  case  if  the  capacity  is  due  to  elec- 
trostatic condensers,  but  not  if  the  capacity  is  that  of  electrolytic 
condensers,  aluminum  cells,  etc. 

Combining  now  the  power-dissipation  equation  (25)  as  factor 
with  the  equations  of  periodic  energy  transfer,  (21)  and  (22), 
gives  the  complete  equations  of  the  double-energy  transient  of 
the  circuit  containing  inductance  and  capacity: 


=  € 


e  =  e 


t  .   t 

cos yQei  sin  -  >  > 

(7  fl- 

CCS  -  +  z0ii  sin     / ' 
o-  °  ) 


(28) 


where 


(29) 


a  =  VLC,  (30) 

and  ii  and  e\  are  the  initial  values  of  the  transient  current  and  volt- 
age respectively. 

As  instance  are  constructed,  in  Fig.  33,  the  transients  of  current 
and  of  voltage  of  a  circuit  having  the  constants  : 


L  =  1.25  mh  =  1.25  X  10~3  henrys; 
C  =  2  mf  =  2  X  lO"6  "farads; 

r  =  2.5  ohms; 

g  =  0.008  mho, 


Inductance, 
Capacity, 
Resistance, 
Conductance, 

in  the  case,  that 

The  initial  transient  current,      ii  =  140  amperes; 
The  initial  transient  voltage,      e\  =  2000  volts. 

It  is,  by  the  preceding  equations: 

a  =  Vie  =  5  x  io-5, 

/  =  -  —  =  3180  cycles  per  second, 

Z  TTff 


ZQ  =  y  ~  =  25  ohms, 

/C 
2/o  =  y  T  =  0.04  mho, 


TO        ELECTRIC  DISCHARGES,   WAVES  AND  IMPULSES. 


Tl  =  —  =  0.001  sec.  =  1  millisecond, 


2C 


=  0.0005  sec.  =  0.5  millisecond, 


1 


0.000333  sec.  =  0.33  millisecond; 


3000  X15& 


/  i  \ 


-1,0 


£1 


,i\i 


\ 


\ 


\ 


Milliseconds 


\ 


Fig.  33. 

hence,  substituted  in  equation  (28), 

i  =  €-3<S140cos0.2£-80sin0.2Zj, 
e  =  e-3<S2000  cos  0.2 1  +  3500  sin  0.2 «}, 

where  the  time  t  is  given  in  milliseconds. 


DOUBLE-ENERGY   TRANSIENTS.  71 

Fig.  33 A  gives  the  periodic  components  of  current  and  voltage: 

if  =  140  cos  0.2 1  -  80  sin  0.2 1, 
er  =  2000  cos  0.2 1  +  3500  sin  0.2 1. 

Fig.  335  gives 

The  magnetic-energy  transient,      h  =  e~', 
The  dielectric-energy  transient,      k  =  e~2<, 
And  the  resultant  transient,          hk  —  e~3<. 

And  Fig.  33(7  gives  the  transient  current,  i  =  hki',  and  the  tran- 
sient voltage,  e  —  like'. 


LECTURE  VII. 
LINE  OSCILLATIONS. 

28.  In  a  circuit  containing  inductance  and  capacity,  the  tran- 
sient consists  of  a  periodic  component,  by  which  the  stored  energy 

7"  /j'2  f^ r/>2 

surges  between  magnetic  -^-  and  dielectric  — ,  and  a  transient 

£i  A 

component,  by  which  the  total  stored  energy  decreases. 

Considering  only  the  periodic  component,  the  maximum  mag- 
netic energy  must  equal  the  maximum  dielectric  energy, 

Lio2  _  Ceo2 
"2"       ~2~' 

where  i0  =  maximum  transient  current,  e0  =  maximum  transient 
voltage. 
This  gives  the  relation  between  eQ  and  io, 

e0      V/L_  1 

i-0  =  \C-ZQ-yQ' 

where  ZQ  is  called  the  natural  impedance  or  surge  impedance,  y0 
the  natural  or  surge  admittance  of  the  circuit. 

As  the  maximum  of  current  must  coincide  with  the  zero  of 
voltage,  and  inversely,  if  the  one  is  represented  by  the  cosine 
function,  the  other  is  the  sine  function;  hence  the  periodic  com- 
ponents of  the  transient  are 

ii  =  IQ  cos  (</>  —  7) 


ei  =  e0  sin  (0  —  7) l 
where 

#  =  2»ft  (4) 

and 

'  =  27^  (5) 

is  the  frequency  of  oscillation. 
The  transient  component  is 

hk  =  e-*,  (6) 

72 


LINE  OSCILLATIONS.  73 

where 


e   =  —  €Q  sin  7 


hence  the  total  expression  of  transient  current  and  voltage  is 
i  =  loe-^cos  (0  -  7) 
6  =  eoe-^sinfa  -  7) 

7,  e0,  and  i.Q  follow  from  the  initial  values  ef  and  i'  of  the  transient, 
at  £  =  Oor  0  =  0: 


hence 


The  preceding  equations  of  the  double-energy  transient  apply 
to  the  circuit  in  which  capacity  and  inductance  are  massed,  as,  for 
instance,  the  discharge  or  charge  of  a  condenser  through  an  in- 
ductive circuit. 

Obviously,  no  material  difference  can  exist,  whether  the  capacity 
and  the  inductance  are  separately  massed,  or  whether  they  are 
intermixed,  a  piece  of  inductance  and  piece  of  capacity  alternating, 
or  uniformly  distributed,  as  in  the  transmission  line,  cable,  etc. 

Thus,  the  same  equations  apply  to  any  point  of  the  transmission 
line. 


A  B 


Fig.  34. 

However,  if  (8)  are  the  equations  of  current  and  voltage  at  a 
point  A  of  a  line,  shown  diagrammatically  in  Fig.  34,  at  any  other 
point  B,  at  distance  I  from  the  point  A,  the  same  equations  will 
apply,  but  the  phase  angle  7,  and  the  maximum  values  eQ  and  IQ, 
may  be  different. 
Thus,  if 

i  =  c<rui  cos  (0  -  7)     )  n  1  x 

sin(0  -7)  ) 


74        ELECTRIC  DISCHARGES,   WAVES  AND  IMPULSES. 

are  the  current  and  voltage  at  the  point  A,  this  oscillation  will 
appear  at  a  point  B,  at  distance  I  from  A,  at  a  moment  of  time 
later  than  at  A  by  the  time  of  propagation  ti  from  A  to  B,  if  the 
oscillation  is  traveling  from  A  to  B-  that  is,  in  the  equation  (11), 
instead  of  t  the  time  (t  —  t\)  enters. 

Or,  if  the  oscillation  travels  from  B  to  A,  it  is  earlier  at  B  by  the 
time  ti'  that  is,  instead  of  the  time  t,  the  value  (t  +  ti)  enters  the 
equation  (11).  In  general,  the  oscillation  at  A  will  appear  at  B, 
and  the  oscillation  at  B  will  appear  at  A,  after  the  time  t\;  that 
is,  both  expressions  of  (11),  with  (t  —  t\)  and  with  (t  +  ti),  will 
occur. 

The  general  form  of  the  line  oscillation  thus  is  given  by  substi- 
tuting (t  T  ti)  instead  of  t  into  the  equations  (11),  where  t\  is  the 
time  of  propagation  over  the  distance  I. 

If  v  =  velocity  of  propagation  of  the  electric  field,  which  in  air, 
as  with  a  transmission  line,  is  approximately 

v  =  3  X  1010,  (12) 

and  in  a  medium  of  permeability  /z  and  permittivity  (specific 

capacity)  K  is 

3  X  1010  (    . 

v  =5  -  T=^>  (13) 

VfUJ 

and  we  denote 

;•;  •  .v  •'.,.     a-j,         ffifil   (14) 

then 

ti  =  al;  (15) 

and  if  we  denote 

co  =  27rM  (16) 


we  get,  substituting  t  =F  t\  for  Z  and  0  =F  co  for  $  into  the  equation 
(11),  the  equations  of  the  line  oscillation: 

i  =  ce~ut  cos  (0  T  co  -  7)       )  ,17, 

6  =  Z0ce-u(  sin  (</>  =F  co  —  7)  ) 

In  these  equations, 

0    =    2  7T/Z         ^ 

is  the  time  angle,  and  (18) 

co  =  2  7r/aZ   ) 

is  the  space  angle,  and  c  is  the  maximum  value  of  current,  ZQC  the 
maximum  value  of  voltage  at  the  point  I. 


LINE  OSCILLATIONS.  75 

Resolving  the  trigonometric  expressions  of  equation  (17)  into 
functions  of  single  angles,  we  get  as  equations  of  current  and  of 
voltage  products  of  the  transient  e~ut,  and  of  a  combination  of  the 
trigonometric  expressions: 

cos  0  cos  co, 
sin  0  cos  co, 
cos  0  sin  co, 
sin  0  sin  co. 

Line  oscillations  thus  can  be  expressed  in  two  different  forms, 
either  as  functions  of  the  sum  and  difference  of  time  angle  0  and 
distance  angle  co:  (0  ±  co),  as  in  (17);  or  as  products  of  functions 
of  0  and  functions  of  co,  as  in  (19).  The  latter  expression  usually 
is  more  convenient  to  introduce  the  terminal  conditions  in  station- 
ary waves,  as  oscillations  and  surges;  the  former  is  often  more 
convenient  to  show  the  relation  to  traveling  waves. 

In  Figs.  35  and  36  are  shown  oscillograms  of  such  line  oscilla- 
tions. Fig.  35  gives  the  oscillation  produced  by  switching  28 
miles  of  100-kv.  line  by  high-tension  switches  onto  a  2500-kw. 
step-up  transformer  in  a  substation  at  the  end  of  a  153-mile  three- 
phase  line;  Fig.  36  the  oscillation  of  the  same  system  caused  by 
switching  on  the  low-tension  side  of  the  step-up  transformer. 

29.  As  seen,  the  phase  of  current  i  and  voltage  e  changes  pro- 
gressively along  the  line  Z,  so  that  at  some  distance  1Q  current  and 
voltage  are  360  degrees  displaced  from  their  values  at  the  starting 
point,  that  is,  are  again  in  the  same  phase.  This  distance  Z0  is 
called  the  wave  length,  and  is  the  distance  which  the  electric  field 

travels  during  one  period  to  =  j  of  the  frequency  of  oscillation. 

As  current  and  voltage  vary  in  phase  progressively  along  the 
line,  the  effect  of  inductance  and  of  capacity,  as  represented  by 
the  inductance  voltage  and  capacity  current,  varies  progressively, 
and  the  resultant  effect  of  inductance  and  capacity,  that  is,  the 
effective  inductance  and  the  effective  capacity  of  the  circuit,  thus 
are  not  the  sum  of  the  inductances  and  capacities  of  all  the  line 
elements,  but  the  resultant  of  the  inductances  and  capacities  of 
all  the  line  elements  combined  in  all  phases.  That  is,  the  effective 

inductance  and  capacity  are  derived  by  multiplying  the  total 

2 
inductance  and  total  capacity  by  avg/cos/,  that  is,  by  -  • 


T6        ELECTRIC  DISCHARGES,   WAVES  AND  IMPULSES. 


LINE  OSCILLATIONS. 


77 


78        ELECTRIC  DISCHARGES,   WAVES  AND  IMPULSES. 
Instead  of  L  and  C,  thus  enter  into  the  equation  of  the  double- 

O    T  Q   H 

energy  oscillation  of  the  line  the  values  -  -  and  — . 

7T  7T 

In  the  same  manner,  instead  of  the  total  resistance  r  and  the 

2  T  2  Q 

total  conductance  g,  the  values  -  —  and  -  -  appear. 

7T  7T 

The  values  of  z0,  y0,  u,  0,  and  co  are  not  changed  hereby. 
The  frequency  /,  however,  changes  from  the  value  correspond- 
ing to  the  circuit  of  massed  capacity,  /  =  -       . ,  to  the  value 

2  IT  VLC 
f  =  4  Vic  * 

Thus  the  frequency  of  oscillation  of  a  transmission  line  is 


where 

(7  =  VLC.  (21) 

If  h  is  the  length  of  the  line,  or  of  that  piece  of  the  line  over 
which  the  oscillation  extends,  and  we  denote  by 

LO,  Co,  TO,  go  (22) 

the  inductance,  capacity,  resistance,  and  conductance  per  unit 

length  of  line,  then 

-i  /  „         ~  \ 

(23) 

that  is,  the  rate  of  decrease  of  the  transient  is  independent  of  the 
length  of  the  line,  and  merely  depends  on  the  line  constants  per 
unit  length. 
It  then  is 

o-  =  Z*ro,  (24) 

where 

—   -\/T    C*  fOf^\ 

(TO   —     *  J-JQ\s  0  \^"/ 

is  a  constant  of  the  line  construction,  but  independent  of  the  length 
of  the  line. 

The  frequency  then  is 

/.-rrr-  (26) 


LINE  OSCILLATIONS.  79 

The  frequency  /  depends  upon  the  length  Zi  of  the  section  of  line 
in  which  the  oscillation  occurs.  That  is,  the  oscillations  occurring 
in  a  transmission  line  or  other  circuit  of  distributed  capacity  have 
no  definite  frequency,  but  any  frequency  may  occur,  depending  on 
the  length  of  the  circuit  section  which  oscillates  (provided  that 
this  circuit  section  is  short  compared  with  the  entire  length  of  the 
circuit,  that  is,  the  frequency  high  compared  with  the  frequency 
which  the  oscillation  would  have  if  the  entire  line  oscillates  as  a 
whole). 

If  Zi  is  the  oscillating  line  section,  the  wave  length  of  this  oscilla- 
tion is  four  times  the  length 

Z0  =  4  ZL  (27) 

This  can  be  seen  as  follows: 

At  any  point  I  of  the  oscillating  line  section  Zi,  the  effective 
power 

Po  =  avg  ei  =  0  (28) 

is  always  zero,  since  voltage  and  current  are  90  degrees  apart. 
The  instantaneous  power 

p  =  ei,  (29) 

however,  is  not  zero,  but  alternately  equal  amounts  of  energy  flow 
first  one  way,  then  the  other  way. 

Across  the  ends  of  the  oscillating  section,  however,  no  energy 
can  flow,  otherwise  the  oscillation  would  not  be  limited  to  this* 
section.  Thus  at  the  two  ends  of  the  section;~the  instantaneous 
power,  and  thus  either  e  or  i,  must  continuously  be  zero. 

Three  cases  thus  are  possible: 

1.  e  =  0  at  both  ends  of  Zx; 

2.  i  =  0  at  both  ends  of  Zi; 

3.  e  =  0  at  one  end,  i  =  0  at  the  other  end  of  Zi. 

In  the  third  case,  i  =  0  at  one  end,  e  =  0  at  the  other  end  of 
the  line  section  Zi,  the  potential  and  current  distribution  in  the 
line  section  Zi  must  be  as  shown  in  Fig.  37,  A,  B,  C,  etc.  That  is, 
Zi  must  be  a  quarter-wave  or  an  odd  multiple  thereof. 

If  Zi  is  a  three-quarters  wave,  in  Fig.  375,  at  the  two  points  C  and 
D  the  power  is  also  zero,  that  is,  Zi  consists  of  three  separate  and 

independent  oscillating  sections,  each  of  the  length  ^  ;  that  is,  the 

o 


80         ELECTRIC  DISCHARGES,    WAVES  AND  IMPULSES. 


unit  of  oscillation  is  -5,  or  also  a  quarter-wave. 
o 


The  same  is  the 


case  in  Fig.  37C,  etc. 

In  the  case  2,  i  =  0  at  both  ends  of  the  line,  the  current  and 
voltage  distribution  are  as  sketched  in  Fig.  38,  A,  B,  C,  etc. 

That  is,  in  A,  the  section  li  is  a  half-wave,  but  the  middle,  C, 
of  li  is  a  node  or  point  of  zero  power,  and  the  oscillating  unit 
again  is  a  quarter-wave.  In  the  same  way,  in  Fig.  385,  the 
section  /i  consists  of  4  quarter- wave  units,  etc. 


Fig.  37. 


Fig.  38. 


The  same  applies  to  case  1,  and  it  thus  follows  that  the  wave 
length  10  is  four  times  the  length  of  the  oscillation  l\. 

30.  Substituting  /0  =  4  li  into  (26)  gives  as  the  frequency  of 
oscillation 

/  =  ^r  •  (30) 

However,  if  /  =  frequency,  and  v  =  - ,  velocity  of  propagation, 


the  wave  length  1Q  is  the  distance  traveled  during  one  period: 

^o  =  -*  =  period, 


(31) 


LINE  OSCILLATIONS.  81 

thus  is 

Zo  =  trfo  =  ^.,  (32) 

and,  substituting  (32)  into  (31),  gives 

a  =  (70,  (33) 

or 

(34) 


This  gives  a  very  important  relation  between  inductance  LO 
and  capacity  Co  per  unit  length,  and  the  velocity  of  propagation. 
It  allows  the  calculation  of  the  capacity  from  the  inductance, 

C0  =  ^  ,  (35) 

and  inversely.  As  in  complex  overhead  structures  the  capacity 
usually  is  difficult  to  calculate,  while  the  inductance  is  easily  de- 
rived, equation  (35)  is  useful  in  calculating  the  capacity  by  means 
of  the  inductance. 

This  equation  (35)  also  allows  the  calculation  of  the  mutual 
capacity,  and  thereby  the  static  induction  between  circuits,  from 
the  mutual  magnetic  inductance. 

The  reverse  equation, 

-  (36) 


is  useful  in  calculating  the  inductance  of  cables  from  their  meas- 
ured capacity,  and  the  velocity  of  propagation  equation  (13). 

31.  If  li  is  the  length  of  a  line,  and  its  two  ends  are  of  different 
electrical  character,  as  the  one  open,  the  other  short-circuited, 
and  thereby  i  =  0  at  one  end,  e  =  0  at  the  other  end,  the  oscilla- 
tion of  this  line  is  a  quarter-wave  or  an  odd  multiple  thereof. 

The  longest  wave  which  may  exist  in  this  circuit  has  the  wave 
length  Z0  =  4  Zi,  and  therefore  the  period  tQ  =  cr0/o  =  4  o-0/i,  that 

is,  the  frequency  /0  =  -A  —r  .     This  is  called  the  fundamental  wave 

4  ooti 
of  oscillation.     In  addition  thereto,  all  its  odd  multiples  can  exist 

as  higher  harmonics,  of  the  respective  wave  lengths  ^  ^  °_  and 
the  frequencies  (2  k  —  1)/0,  where  k  =  1,  2,  3  .  .  . 


82        ELECTRIC  DISCHARGES,   WAVES  AND  IMPULSES. 

If  then  0  denotes  the  time  angle  and  co  the  distance  angle  of  the 
fundamental  wave,  that  is,  0  =  2  TT  represents  a  complete  cycle 
and  co  =  2ir  a  complete  wave  length  of  the  fundamental  wave, 
the  time  and  distance  angles  of  the  higher  harmonics  are 

30,  3  co, 
50,  5  co, 
70,  7  co,  etc. 

A  complex  oscillation,  comprising  waves  of  all  possible  fre- 
quencies, thus  would  have  the  form 

«i  cos  (0  =F  co  —  71)  +  a3  cos  3  (0  =F  co  —  73) 

+  a5  cos  5  (0  T  co  -  75)  +  .  .  .  ,  (37) 

and  the  length  h  of  the  line  then  is  represented  by  the  angle 
co  =  ~,  and  the  oscillation  called  a  quarter-wave  oscillation. 

If  the  two  ends  of  the  line  h  have  the  same  electrical  charac- 
teristics, that  is,  e  =  0  at  both  ends,  or  i  =  0,  the  longest  possible 
wave  has  the  length  10  =  2  l\,  and  the  frequency 

r     1      1 

J°  ~       T  ~"  o      T  ' 

" 


or  any  multiple  (odd  or  even)  thereof. 

If  then  0  and  co  again  represent  the  time  and  the  distance 
angles  of  the  fundamental  wave,  its  harmonics  have  the  respective 
time  and  distance  angles 

20,  2  co, 
30,  3  co, 
40,  4  co,  etc. 

A  complex  oscillation  then  has  the  form 

a\  cos  (0  =F  co  —  71)  +  «2  cos  2  (0  T  co  —  72) 

+  a3  cos  3  (0  =F  co  -  73)  +  .  .  .  ,  (38) 

and  the  length  l\  of  the  line  is  represented  by  angle  coi  =  TT,  and  the 
oscillation  is  called  a  half  -wave  oscillation. 

The  half-wave  oscillation  thus  contains  even  as  well  as  odd 
harmonics,  and  thereby  may  have  a  wave  shape,  in  which  one 
half  wave  differs  from  the  other. 

Equations  (37)  and  (38)  are  of  the  form  of  equation  (17),  but 


LINE  OSCILLATIONS.  83 

usually  are  more  conveniently  resolved  into  the  form  oi  equa- 
tion (19). 

At  extremely  high  frequencies  (2  k  —I)/,  that  is,  for  very  large 
values  of  k,  the  successive  harmonics  are  so  close  together  that  a 
very  small  variation  of  the  line  constants  causes  them  to  overlap, 
and  as  the  line  constants  are  not  perfectly  constant,  but  may 
vary  slightly  with  the  voltage,  current,  etc.,  it  follows  that  at  very 
high  frequencies  the  line  responds  to  any  frequency,  has  no  definite 
frequency  of  oscillation,  but  oscillations  can  exist  of  any  frequency, 
provided  this  frequency  is  sufficiently  high.  Thus  in  transmission 
lines,  resonance  phenomena  can  occur  only  with  moderate  frequen- 
cies, but  not  with  frequencies  of  hundred  thousands  or  millions  of 
cycles. 

32.  The  line  constants  r0,  go,  L0,  C0  are  given  per  unit  length, 
as  per  cm.,  mile,  1000  feet,  etc. 

The  most  convenient  unit  of  length,  when  dealing  with  tran- 
sients in  circuits  of  distributed  capacity,  is  the  velocity  unit  v. 

That  is,  choosing  as  unit  of  length  the  distance  of  propagation 
in  unit  time,  or  3  X  1010  cm.  in  overhead  circuits,  this  gives  v  =  1, 
and  therefore 


"- 

T         1 
or  GO  —  -j-  ;  LIQ  —  -ftj- 


1 
j 

-L/o 


That  is,  the  capacity  per  unit  of  length,  in  velocity  measure,  is 
inversely  proportional  to  the  inductance.  In  this  velocity  unit  of 
length,  distances  will  be  represented  by  X. 

Using  this  unit  of  length,  <70  disappears  from  the  equations  of 
the  transient. 

This  velocity  unit  of  length  becomes  specially  useful  if  the 
transient  extends  over  different  circuit  sections,  of  different  con- 
stants and  therefore  different  wave  lengths,  as  for  instance  an 
overhead  line,  the  underground  cable,  in  which  the  wave  length  is 
about  one-half  what  it  is  in  the  overhead  line  (K  =  4)  and  coiled 
windings,  as  the  high-potential  winding  of  a  transformer,  in  which 
the  wave  length  usually  is  relatively  short.  In  the  velocity 
measure  of  length,  the  wave  length  becomes  the  same  throughout 
all  these  circuit  sections,  and  the  investigation  is  thereby  greatly 
simplified. 


84        ELECTRIC  DISCHARGES,   WAVES  AND  IMPULSES. 
Substituting  O-Q  =  1  in  equations  (30)  and  (31)  gives 
^o  =  Ao> 


O     ^          27rX 

CO   =   2  7T/X   =    — —  ; 

AO 


(40) 


and  the  natural  impedance  of  the  line  then  becomes,  in  velocity 
measure, 

4    /  LQ  T  1  1  ^O  /A1\ 

z° =  V  r  =  L° =  T  =  ?T  =  T  (41) 

^o  ^o       2/o       ^o 

where  e0  =  maximum  voltage,  i0  =  maximum  current. 

That  is,  the  natural  impedance  is  the  inductance,  and  the 
natural  admittance  is  the  capacity,  per  velocity  unit  of  length, 
and  is  the  main  characteristic  constant  of  the  line. 

The  equations  of  the  current  and  voltage  of  the  line  oscillation 
then  consist,  by  (19),  of  trigonometric  terms 

cos  0  cos  co, 
sin  0  cos  cu, 
cos  0  sin  co, 
sin  <f>  sin  co, 

multiplied  with  the  transient,  e~ut,  and  would  thus,  in  the  most 
general  case,  be  given  by  an  expression  of  the  form 

i  =  e~  "*  I  ai  cos  </>  cos  co  +  61  sin  0  cos  co  +  Ci  cos  0  sin  co 

-f-disin</>sinco|, 
e  _  €-  ut  |  fll/  cos  ^  cos  w  _|_  ^  sm  ^  cos  w  _j_  Cl/  cos  ^  sm  w 

+  di  sin  0  sin  co  j , 

and  its  higher  harmonics,  that  is,  terms,  with 

20,  2  co, 
30,  3  co, 
40,  4  co,  etc. 

In  these  equations  (42),  the  coefficients  a,  6,  c,  d,  a',  6',  c',  d' 
are  determined  by  the  terminal  conditions  of  the  problem,  that 
is,  by  the  values  of  i  and  e  at  all  points  of  the  circuit  co,  at  the 


LINE  OSCILLATIONS.  85 

beginning  of  time,  that  is,  for  <£  =  0,  and  by  the  values  of  i  and  e 
at  all  times  t  (or  </>  respectively)  at  the  ends  of  the  circuit,  that  is, 

for  co  =  0  and  co  =  •=• 
ft 

For  instance,  if: 

(a)  The  circuit  is  open  at  one  end  co  =  0,  that  is,  the  current 
is  zero  at  all  times  at  this  end.  That  is, 

i  =  0  for  co  =  0; 

the  equations  of  i  then  must  not  contain  the  terms  with  cos  co, 
cos  2  co,  etc.,  as  these  would  not  be  zero  for  co  =  0.  That  is,  it 
must  be 

Ol  ==  0,     61  =  0,  ) 

a2  =  0,     62  =  0,  (43) 

a3  =  0,     63  =  0,  etc.    ) 

The  equation  of  i  contains  only  the  terms  with  sin  co,  sin  2  co, 
etc.  Since,  however,  the  voltage  e  is  a  maximum  where  the 
current  i  is  zero,  and  inversely,  at  the  point  where  the  current  is 
zero,  the  voltage  must  be  a  maximum;  that  is,  the  equations  of 
the  voltage  must  contain  only  the  terms  with  cos  co,  cos  2  co,  etc. 
Thus  it  must  be 

ci'  =  0,     d/  =  0,          ) 

C2'  =  0,     cV  =  0,  (44) 

c8'  =  0,     d8'  =  0,  etc.J 

Substituting  (43)  and  (44)  into  (42)  gives 

i  =  c~ui  \d  cos  </>  +  di  sin  <£ 1  sin  co,     |  ,     . 

e  =  e~ut  {ai  cos  0  +  bi  sin  0}  cos  co    ) 

and  the  higher  harmonics  hereof. 

(6)  If  in  addition  to  (a),  the  open  circuit  at  one  end  co  =  0, 

7T 

the  line  is  short-circuited  at  the  other  end  co  =  -,  the  voltage  e 

a 

must  be  zero  at  this  latter  end.  Cos  co,  cos  3  co,  cos  5  co,  etc., 
become  zero  for  co  =  »,  but  cos  2  co,  cos  4  co,  etc.,  are  not  zero  for 

7T 

co  =  ^,  and  the  latter  functions  thus  cannot  appear  in  the  expres- 
sion of  e. 


86        ELECTRIC  DISCHARGES,   WAVES  AND  IMPULSES. 

That  is,  the  voltage  e  can  contain  no  even  harmonics.  If, 
however,  the  voltage  contains  no  even  harmonics,  the  current 
produced  by  this  voltage  also  can  contain  no  even  harmonics. 
That  is,  it  must  be 

C2  =  0,     ^  =  0,     a/  =  0,     62'  =  0,  ) 

c4  =  0,     d,  =  0,     a,'  =  0,     64'  =  0,  (46) 

C6  =  0,     d6  =  0,     a6'  =  0,     66'  =  0,    etc.    ) 

The  complete  expression  of  the  stationary  oscillation  in  a  circuit 
open  at  the  end  co  =  0  and  short-circuited  at  the  end  co  =  ^  thus 
would  be 

i  =  e~ut  I  (ci  cos  0  +  di  sin  0)  sin  co  +  (c3  cos  3  0  +  d3  sin  3  0) 

sin3co  +  .  .  .   j, 
e  =  e~ut  I  (a/  cos  0  +  bi  sin  0)  cos  co  +  (a/  cos  3  0  +  63'  sin  3  0) 

cos  3  w  +...}. 

(c)  Assuming  now  as  instance  that,  in  such  a  stationary  oscilla- 
tion as  given  by  equation  (47),  the  current  in  the  circuit  is  zero 
at  the  starting  moment  of  the  transient  for  0  =  0.  Then  the 
equation  of  the  current  can  contain  no  terms  with  cos  0,  as  these 
would  not  be  zero  for  0  =  0. 

That  is,  it  must  be 

c3  =  0,          [  (48) 

c5  =  0,  etc.  ) 

At  the  moment,  however,  when  the  current  is  zero,  the  voltage 
of  the  stationary  oscillation  must  be  a  maximum.  As  i  =  0  for 
0  =  0,  at  this  moment  the  voltage  e  must  be  a  maximum,  that 
is,  the  voltage  wave  can  contain  no  terms  with  sin  0,  sin  3  0,  etc. 
This  means 

V  =  0,          ) 

63'  =  0,  (49) 

65'  =  0,  etc.  ) 

Substituting  (48)  and  (49)  into  equation  (47)  gives 

sin  0  sin  co  +  d3  sin  3  0  sin  3  co  +  c?5  sin  5  0  sin  5  co 

(50) 


e  =  «-"'  \ai  cos  <t>  cos  w+O)  cos  3  0  cos  3  w+o5'  cos  5  <t>  cos  5  w 
+  .  .  .   1. 


LINE  OSCILLATIONS.  87 

In  these  equations  (50),  d  and  a'  are  the  maximum  values  of 
current  and  of  voltage  respectively,  of  the  different  harmonic 
waves.  Between  the  maximum  values  of  current,  i0,  and  of  volt- 
age, eo,  of  a  stationary  oscillation  exists,  however,  the  relation 


where  z0  is  the  natural  impedance  or  surge  impedance.     That  is 

a'=dz0)  (51) 

and  substituting  (51)  into  (50)  gives 
i  —  e~ut  \  di  sin  <f>  sin  co  +  d$  sin  3  <f>  sin  3  co  +  d$  sin  5  <j>  sin  5  co 


e  =  z0  €~ut  I  di  cos  0  cos  co  +  d3  cos  3  <£  cos  3  co  -j-  d5  cos  5  0  cos  5 


(52) 


(d)  If  then  the  distribution  of  voltage  e  along  the  circuit  is  given 
at  the  moment  of  start  of  the  transient,  for  instance,  the  voltage 
is  constant  and  equals  e\  throughout  the  entire  circuit  at  the 
starting  moment  0  =  0  of  the  transient,  this  gives  the  relation, 
by  substituting  in  (52), 

ei  =  ZQ  t~ut  \  di  cos  co  +  c?3  cos  3  co  +  c?5  cos  5  co  +  .  .  .   } ,     (53) 

for  all  values  of  co. 

Herefrom  then  calculate  the  values  of  d\,  d3,  d$,  etc.,  in  the 
manner  as  discussed  in  "  Engineering  Mathematics,"  Chapter  III. 


LECTURE  VIII. 
TRAVELING   WAVES. 

33.  In  a  stationary  oscillation  of  a  circuit  having  uniformly 
distributed  capacity  and  inductance,  that  is,  the  transient  of  a 
circuit  storing  energy  in  the  dielectric  and  magnetic  field,  current 
and  voltage  are  given  ^by  the  expression 

i  =  iQe~ut  cos  (0  T  co  -  7),  ) 
e  =  e0e~ut  sin  (</>  T  co  —  7),  ) 

where  0  is  the  time  angle,  co  the  distance  angle,  u  the  exponential 
decrement,  or  the  "power-dissipation  constant,"  and  i0  and  eQ  the 
maximunl  current  and  voltage  respectively. 

The  power  flow  at  any  point  of  the  circuit,  that  is,  at  any  dis- 
tance angle  co,  and  at  any  time  t,  that  is,  time  angle  <£,  then  is 

p  =  ei, 

=  e0ioe~2ut  cos  (</>  T  co  —  7)  sin  (0  =F  co  —  7), 

=  ^|V2«<sin2(c/>=Fco-7),  (2) 

and  the  average  power  flow  is 

Po  =  avg  p,  (3) 

=  0. 

Hence,  in  a  stationary  oscillation,  or  standing  wave  of  a  uni- 
form circuit,  the  average  flow  of  power,  p0,  is  zero,  and  no  power 
flows  along  the  circuit,  but  there  is  a  surge  of  power,  of  double 
frequency.  That  is,  power  flows  first  one  way,  during  one-quarter 
cycle,  and  then  in  the  opposite  direction,  during  the  next  quarter- 
cycle,  etc. 

Such  a  transient  wave  thus  is  analogous  to  the  permanent  wave 
of  reactive  power. 

As  in  a  stationary  wave,  current  and  voltage  are  in  quadrature 
with  each  other,  the  question  then  arises,  whether,  and  what 


TRAVELING  WAVES. 


89 


physical  meaning  a  wave  has,  in  which  current  and  voltage  are  in 
phase  with  each  other: 

i  =  loe~ut  COS  (0  =F  co  —  7), 
e  =  eQe~ut  cos  (<£  =F  «  —  7). 

In  this  case  the  flow  of  power  is 


(4) 


P  = 


=  eQiQe-2ut  cos2 


co  -  7), 


and  the  average  flow  of  power  is 

p0  =  avg  p, 


(5) 


(6) 


Such  a  wave  thus  consists  of  a  combination  of  a  steady  flow  of 
power  along  the  circuit,  p0)  and  a  pulsation  or  surge,  pi,  of  the  same 
nature  as  that  of  the  standing  wave  (2) : 

Such  a  flow  of  power  along  the  circuit  is  called  a  traveling  wave. 
It  occurs  very  frequently.  For  instance,  it  may  be  caused  if  by  a 
lightning  stroke,  etc.,  a  quantity  of  dielectric  energy  is  impressed 


A       • 


Fig.  39.  —  Starting  of  Impulse,  or  Traveling  Wave. 

upon  a  part  of  the  circuit,  as  shown  by  curve  A  in  Fig.  39,  or  if  by  a 
local  short  circuit  a  quantity  of  magnetic  energy  is  impressed  upon 
a  part  of  the  circuit.  This  energy  then  gradually  distributes  over 
the  circuit,  as  indicated  by  the  curves  B,  C,  etc.,  of  Fig.  39,  that  is, 
moves  along  the  circuit,  and  the  dissipation  of  the  stored  energy 
thus  occurs  by  a  flow  of  power  along  the  circuit. 


90        ELECTRIC  DISCHARGES,   WAVES  AND  IMPULSES. 

Such  a  flow  of  power  must  occur  in  a  circuit  containing  sections 
of  different  dissipation  constants  u.  For  instance,  if  a  circuit 
consists  of  an  unloaded  transformer  and  a  transmission  line,  as 
indicated  in  Fig.  40,  that  is,  at  no  load  on  the  step-down  trans- 

^>  Line 


Transformer 
Line 


Fig.   40. 

former,  the  high-tension  switches  are  opened  at  the  generator 
end  of  the  transmission  line.  The  energy  stored  magnetically  and 
dielectrically  in  line  and  transformer  then  dissipates  by  a  transient, 
as  shown  in  the  oscillogram  Fig.  41.  This  gives  the  oscillation 
of  a  circuit  consisting  of  28  miles  of  line  and  2500-kw.  100-kv. 
step-up  and  step-down  transformers,  and  is  produced  by  discon- 
necting this  circuit  by  low-tension  switches.  In  the  transformer, 
the  duration  of  the  transient  would  be  very  great,  possibly  several 
seconds,  as  the  stored  magnetic  energy  (L)  is  very  large,  the  dis- 
sipation of  power  (r  and  g)  relatively  small;  in  the  line,  the  tran- 
sient is  of  fairly  short  duration,  as  r  (and  g)  are  considerable. 
Left  to  themselves,  the  line  oscillations  thus  would  die  out  much 
more  rapidly,  by  the  dissipation  of  their  stored  energy,  than  the 
transformer  oscillations.  Since  line  and  transformer  are  connected 
together,  both  must  die  down  simultaneously  by  the  same  tran- 
sient. It  then  follows  that  power  must  flow  during  the  transient 
from  the  transformer  into  the  line,  so  as  to  have  both  die  down 
together,  in  spite  of  the  more  rapid  energy  dissipation  in  the  line. 
Thus  a  transient  in  a  compound  circuit,  that  is,  a  circuit  comprising 
sections  of  different  constants,  must  be  a  traveling  wave,  that  is, 
must  be  accompanied  by  power  transfer  between  the  sections  of 
the  circuit.* 

A  traveling  wave,  equation  (4),  would  correspond  to  the  case  of 
effective  power  in  a  permanent  alternating-current  circuit,  while 
the  stationary  wave  of  the  uniform  circuit  corresponds  to  the  case 
of  reactive  power. 

Since  one  of*the  most  important  applications  of  the  traveling 
wave  is  the  investigation  of  the  compound  circuit,  it  is  desirable 

*  In  oscillogram  Fig.  41,  the  current  wave  is  shown  reversed  with  regard 
to  the  voltage  wave  for  greater  clearness. 


TRAVELING  WAVES. 


91 


92        ELECTRIC  DISCHARGES,   WAVES  AND  IMPULSES. 


to  introduce,  when  dealing  with  traveling  waves,  the  velocity  unit 
as  unit  of  length,  that  is,  measure  the  length  with  the  distance  of 
propagation  during  unit  time  (3  X  1010  cm.  with  a  straight  con- 
ductor in  air)  as  unit  of  length.  This  allows  the  use  of  the  same 
distance  unit  through  all  sections  of  the  circuit,  and  expresses  the 
wave  length  X0  and  the  period  T0  by  the  same  numerical  values, 

X0  =  TQ  =  -,  and  makes  the  time  angle  0  and  the  distance  angle  co 
directly  comparable: 


0    =    2vft    =    27T— , 

AO 

CO    =    2  7T  —   =    2  7T/X. 
A 


(8) 


34.   If  power  flows  along  the  circuit,  three  cases  may  occur: 
(a)  The  flow  of  power  is  uniform,  that  is,  the  power  remains 

constant  in  the  direction  of  propagation,  as  indicated  by  A  in 

Fig.  42. 


B 


B 

c' 

A' 
B' 


Fig.  42.  —  Energy  Transfer  by  Traveling  Wave. 

(b)  The  flow  of  power  is  decreasing  in  the  direction  of  propaga- 
tion, as  illustrated  by  B  in  Fig.  42. 

(c)  The  flow  of  power  is  increasing  in  the  direction  of  propaga- 
tion, as  illustrated  by  C  in  Fig.  42. 

Obviously,  in  all  three  cases  the  flow  of  power  decreases,  due  to 
the  energy  dissipation  by  r  and  g,  that  is,  by  the  decrement  e~ut. 
Thus,  in  case  (a)  the  flow  of  power  along  the  circuit  decreases  at 


TRAVELING  WAVES.  93 

the  rate  e~ut,  corresponding  to  the  dissipation  of  the  stored  energy 
by  e-"',  as  indicated  by  A  '  in  Fig.  42;  while  in  the  case  (6)  the 
power  flow  decreases  faster,  in  case  (c)  slower,  than  corresponds 
to  the  energy  dissipation,  and  is  illustrated  by  B'  and  C'  in  Fig.  42. 
(a)  If  the  flow  of  power  is  constant  in  the  direction  of  propa- 
gation, the  equation  would  be 

i  =  io<rut  cos  (0  —  o>  —  7), 

e  =  e^~ut  cos  (0  -  co  -  7),  (9) 


In  this  case  there  must  be  a  continuous  power  supply  at  the 
one  end,  and  power  abstraction  at  the  other  end,  of  the  circuit 
or  circuit  section  in  which  the  flow  of  power  is  constant.  This 
could  occur  approximately  only  in  special  cases,  as  in  a  circuit 
section  of  medium  rate  of  power  dissipation,  u,  connected  between 
a  section  of  low-  and  a  section  of  high-power  dissipation.  For 
instance,  if  as  illustrated  in  Fig.  43  we  have  a  transmission  line 

Line 


Transformer  LoadCT 

Line ^-) 

Fig.  43.  —  Compound  Circuit. 

connecting  the  step-up  transformer  with  a  load  on  the  step-down 
end,  and  the  step-up  transformer  is  disconnected  from  the  gener- 
ating system,  leaving  the  system  of  step-up  transformer,  line,  and 
load  to  die  down  together  in  a  stationary  oscillation  of  a  compound 
circuit,  the  rate  of  power  dissipation  in  the  transformer  then 
is  much  lower,  and  that  in  the  load  may  be  greater,  than  the 
average  rate  of  power  dissipation  of  the  system,  and  the  trans- 
former will  supply  power  to  the  rest  of  the  oscillating  system,  the 
load  receive  power.  If  then  the  rate  of  power  dissipation  of  the 
line  u  should  happen  to  be  exactly  the  average,  w0,  of  the  entire 
system,  power  would  flow  from  the  transformer  over  the  line  into 
the  load,  but  in  the  line  the  flow  of  power  would  be  uniform,  as 
the  line  neither  receives  energy  from  nor  gives  off  energy  to  the 
rest  of  the  system,  but  its  stored  energy  corresponds  to  its  rate 
of  power  dissipation. 


94         ELECTRIC  DISCHARGES,   WAVES  AND  IMPULSES. 

(b)  If  the  flow  of  power  decreases  along  the  line,  every  line 
element  receives  more  power  at  one  end  than  it  gives  off  at  the 
other  end.  That  is,  energy  is  supplied  to  the  line  elements  by 
the  flow  of  power,  and  the  stored  energy  of  the  line  element  thus 
decreases  at  a  slower  rate  than  corresponds  to  its  power  dissipation 
by  r  and  g.  Or,  in  other  words,  a  part  of  the  power  dissipated  in 
the  line  element  is  supplied  by  the  flow  of  power  along  the  line, 
and  only  a  part  supplied  by  the  stored  energy. 

Since  the  current  and  voltage  would  decrease  by  the  term  e~w<, 
if  the  line  element  had  only  its  own  stored  energy  available,  when 
receiving  energy  from  the  power  flow  the  decrease  of  current  and 
voltage  would  be  slower,  that  is,  by  a  term 


hence  the  exponential  decrement  u  is  decreased  to  (u  —  s),  and  s 
then  is  the  exponential  coefficient  corresponding  to  the  energy 
supply  by  the  flow  of  power. 

Thus,  while  u  is  called  the  dissipation  constant  of  the  circuit,  s 
may  be  called  the  power-transfer  constant  of  the  circuit. 

Inversely,  however,  in  its  propagation  along  the  circuit,  X,  such 
a  traveling  wave  must  decrease  in  intensity  more  rapidly  than 
corresponds  to  its  power  dissipation,  by  the  same  factor  by  which 
it  increased  the  energy  supply  of  the  line  elements  over  which  it 
passed.  That  is,  as  function  of  the  distance,  the  factor  e~  sX  must 
enter.*  In  other  words,  such  a  traveling  wave,  in  passing  along 
the  line,  leaves  energy  behind  in  the  line  elements,  at  the  rate 
e  +  st,  and  therefore  decreases  faster  in  the  direction  of  progress 
by  e~  sX.  That  is,  it  scatters  a  part  of  its  energy  along  its  path 
of  travel,  and  thus  dies  down  more  rapidly  with  the  distance  of 
travel. 

Thus,  in  a  traveling  wave  of  decreasing  power  flow,  the  time 
decrement  is  changed  to  e~(u~s^,  and  the  distance  decrement  e+sX 
added,  and  the  equation  of  a  traveling  wave  of  decreasing  power 
flow  thus  is 

---  ( 

( 

*  Due  to  the  use  of  the  velocity  unit  of  length  X,  distance  and  time  are 
given  the  same  units,  ^  =  X0;  and  the  time  decrement,  e+*<,  and  the  distance 
decrement,  e~sX,  give  the  same  coefficient  s  in  the  exponent.  Otherwise,  the 
velocity  of  propagation  would  enter  as  factor  in  the  exponent. 


TRAVELING   WAVES.  95 

the  average  power  then  is 


Po  =  avg  e, 


-s)t  e-2s\  —          L°  €-2ute+2s(t-\)  ^ 

2 


Both  forms  of  the  expressions  of  i,  e,  and  po  of  equations  (11) 
and  (12)  are  of  use.  The  first  form  shows  that  the  wave  de- 
creases slower  with  the  time  t,  but  decreases  with  the  distance  X. 
The  second  form  shows  that  the  distance  X  enters  the  equation 
only  in  the  form  t  —  X  and  4>  —  co  respectively,  and  that  thus  for 
a  constant  value  of  t  —  \  the  decrement  is  e~2ut}  that  is,  in  the 
direction  of  propagation  the  energy  dies  out  by  the  power  dissi- 
pation constant  u. 

Equations  (10)  to  (12)  apply  to  the  case,  when  the  direction 
of  propagation,  that  is,  of  wave  travel,  is  toward  increasing  X. 
For  a  wave  traveling  in  opposite  direction,  the  sign  of  X  and  thus 
of  co  is  reversed. 

(c)  If  the  flow  of  power  increases  along  the  line,  more  power 
leaves  every  line  element  than  enters  it;  that  is,  the  line  element 
is  drained  of  its  stored  energy  by  the  passage  of  the  wave,  and  thus 
the  transient  dies  down  with  the  time  at  a  greater  rate  than  corre- 
sponds to  the  power  dissipation  by  r  and  g.  That  is,  not  all  the 
stored  energy  of  the  line  elements  supplies  the  power  which  is 
being  dissipated  in  the  line  element,  but  a  part  of  the  energy 
leaves  the  line  element  in  increasing  the  power  which  flows  along 
the  line.  The  rate  of  dissipation  thus  is  increased,  and  instead 
of  u,  (u  +  s)  enters  the  equation.  That  is,  the  exponential  time 
decrement  is 

e~  <"  +  •)',  (13) 

but  inversely,  along  the  line  X  the  power  flow  increases,  that  is, 
the  intensity  of  the  wave  increases,  by  the  same  factor  e+sX,  or 
rather,  the  wave  decreases  along  the  line  at  a  slower  rate  than 
corresponds  to  the  power  dissipation. 
The  equations  then  become: 

-u<-s^-x)COs(0-co-7),   ) 
6-s(t-x)  cos  <>  —  a  —  * 


and  the  average  power  is 


96        ELECTRIC  DISCHARGES,   WAVES  AND  IMPULSES. 

that  is,  the  power  decreases  with  the  time  at  a  greater,  but  with 
the  distance  at  a  slower,  rate  than  corresponds  to  the  power 
dissipation. 

For  a  wave  moving  in  opposite  direction,  again  the  sign  of  X 
and  thus  of  co  would  be  reversed. 

35.  In  the  equations  (10)  to  (15),  the  power-transfer  constant 
s  is  assumed  as  positive.  In  general,  it  is  more  convenient  to 
assume  that  s  may  be  positive  or  negative;  positive  for  an  increas- 
ing, negative  for  a  decreasing,  flow  of  power.  The  equations  (13) 
to  (15)  then  apply  also  to  the  case  (6)  of  decreasing  power  flow, 
but  in  the  latter  case  s  is  negative.  They  also  apply  to  the  case 
(a)  for  s  =  0. 

The  equation  of  current,  voltage,  and  power  of  a  traveling  wave 
then  can  be  combined  in  one  expression: 


i  =  '^ _       _  ^ 

Q  =  ^£~VW-r*Vc=csAr»rka  i  ^3^,.,  —  /v  }  ==/?„£— at  £—s  u-r  A;   r»r\c  f  .^^ /.i  —  o/ 1     I      V§W 


where  the  upper  sign  applies  to  a  wave  traveling  in  the  direction 
toward  rising  values  of  X,  the  lower  sign  to  a  wave  traveling  in 
opposite  direction,  toward  decreasing  X.  Usually,  waves  of  both 
directions  of  travel  exist  simultaneously  (and  in  proportions  de- 
pending on  the  terminal  conditions  of  the  oscillating  system,  as 
the  values  of  i  and  e  at  its  ends,  etc.). 

s  =  0  corresponds  to  a  traveling  wave  of  constant  power  flow 
(case  (a)). 

s  >  0  corresponds  to  a  traveling  wave  of  increasing  power  flow, 
that  is,  a  wave  which  drains  the  circuit  over  which  it  travels  of 
some  of  its  stored  energy,  and  thereby  increases  the  time  rate  of 
dying  out  (case  (c)). 

s  <  0  corresponds  to  a  traveling  wave  of  decreasing  power  flow, 
that  is,  a  wave  which  supplies  energy  to  the  circuit  over  which  it 
travels,  and  thereby  decreases  the  time  rate  of  dying  out  of  the 
transient. 

If  s  is  negative,  for  a  transient  wave,  it  always  must  be 


since,  if  —  s  >  u,  u  -\-  s  would  be  negative,  and  e~(u  +  s}t  would 
increase  with  the  time;  that  is,  the  intensity  of  the  transient  would 


TRAVELING  WAVES.  97 

increase  with  the  time,  which  in  general  is  not  possible,  as  the 
transient  must  decrease  with  the  time,  by  the  power  dissipation 
in  r  and  g. 

Standing  waves  and  traveling  waves,  in  which  the  coefficient 
in  the  exponent  of  the  time  exponential  is  positive,  that  is,  the 
wave  increases  with  the  time,  may,  however,  occur  in  electric  cir- 
cuits in  which  the  wave  is  supplied  with  energy  from  some  outside 
source,  as  by  a  generating  system  flexibly  connected  (electrically) 
through  an  arc.  Such  waves  then  are  "cumulative  oscillations." 
They  may  either  increase  in  intensity  indefinitely,  that  is,  up  to 
destruction  of  the  circuit  insulation,  or  limit  themselves  by  the 
power  dissipation  increasing  with  the  increasing  intensity  of  the 
oscillation,  until  it  becomes  equal  to  the  power  supply.  Such 
oscillations,  which  frequently  are  most  destructive  ones,  are  met  in 
electric  systems  as  "arcing  grounds,"  "grounded  phase,"  etc. 
They  are  frequently  called  "undamped  oscillations,"  and  as  such 
find  a  use  in  wireless  telegraphy  and  telephony.  Thus  far,  the 
only  source  of  cumulative  oscillation  seems  to  be  an  energy  supply 
over  an  arc,  especially  an  unstable  arc.  In  the  self-limiting  cumu- 
lative oscillation,  the  so-called  damped  oscillation,  the  transient 
becomes  a  permanent  phenomenon.  Our  theoretical  knowledge  of 
the  cumulative  oscillations  thus  far  is  rather  limited,  however. 

An  oscillogram  of  a  "grounded  phase  "  on  a  154-mile  three- 
phase  line,  at  82  kilovolts,  is  given  in  Figs.  44  and  45.  Fig.  44 
shows  current  and  voltage  at  the  moment  of  formation  of  the 
ground;  Fig.  45  the  same  one  minute  later,  when  the  ground  was 
fully  developed. 

An  oscillogram  of  a  cumulative  oscillation  in  a  2500-kw.  100,000- 
volt  power  transformer  (60-cycle  system)  is  given  in  Fig.  46.  It 
is  caused  by  switching  off  28  miles  of  line  by  high-tension  switches, 
at  88  kilovolts.  As  seen,  the  oscillation  rapidly  increases  in  in- 
tensity, until  it  stops  by  the  arc  extinguishing,  or  by  the  destruc- 
tion of  the  transformer. 

Of  special  interest  is  the  limiting  case, 

—  s  =  u; 

in  this  case,  u  +  s  =  0,  and  the  exponential  function  of  time 
vanishes,  and  current  and  voltage  become 

i  =  i0e±sX  cos  (0  =F  co  —  7), 


e  =  e0e±sX  cos  (0  T  co  -  7),  v 


98        ELECTRIC  DISCHARGES,   WAVES  AND  IMPULSES. 


8 


TRAVELING  WAVES. 


99 


100      ELECTRIC  DISCHARGES,   WAVES  AND  IMPULSES. 

that  is,  are  not  transient,  but  permanent  or  alternating  currents 
and  voltages. 

Writing  the  two  waves  in  (18)  separately  gives 


cos  (0  -  co  -  70  -  i'0'e-sX 


e  =  e0e+sx  cos  (0  -  co  - 


and  these  are  the  equations  of  the  alternating-current  transmission 
line,  and  reduce,  by  the  substitution  of  the  complex  quantity  for 
the  function  of  the  time  angle  <f>,  to  the  standard  form  given  in 
"Transient  Phenomena,"  Section  III. 

36.  Obviously,  traveling  waves  and  standing  waves  may  occur 
simultaneously  in  the  same  circuit,  and  usually  do  so,  just  as  in 
alternating-current  circuits  effective  and  reactive  waves  occur 
simultaneously.  In  an  alternating-current  circuit,  that  is,  in 
permanent  condition,  the  wave  of  effective  power  (current  in 
phase  with  the  voltage)  and  'the  wave  of  reactive  power  (current 
in  quadrature  with  the  voltage)  are  combined  into  a  single  wave, 
in  which  the  current  is  displaced  from  the  voltage  by  more  than  0 
but  less  than  90  degrees.  This  cannot  be  done  with  transient 
waves.  The  transient  wave  of  effective  power,  that  is,  the  travel- 
ing wave, 

i  =  iQ€-  ut  €-  s  (t  ±\)  cos  (^  =p  w  _  T)? 

e  =  eQ€~  ut  e~  s  (i  ±X)  cos  (0  =F  co  —  7), 

cannot  be  combined  with  the  transient  wave  of  reactive  power, 
that  is,  the  stationary  wave, 

i  =  io'e-ut  cos  (0  T  co  -  7'), 
e  =  e0'e-ut  sin  (<£  =F  co  -  7'), 

to  form  a  transient  wave,  in  which  current  and  voltage  differ  in 
phase  by  more  than  0  but  less  than  90  degrees,  since  the  traveling 
wave  contains  the  factor  e-s«TX),  resulting  from  its  progression 
along  the  circuit,  while  the  stationary  wave  does  not  contain  this 
factor,  as  it  does  not  progress. 

This  makes  the  theory  of  transient  waves  more  complex  than 
that  of  alternating  waves. 

Thus  traveling  waves  and  standing  waves  can  be  combined  only 
locally,  that  is,  the  resultant  gives  a  wave  in  which  the  phase  angle 
between  current  and  voltage  changes  with  the  distance  X  and  with 
the  time  t. 


TRAVELING  WAVES. 


101 


When  traveling  waves  and  stationary  waves  occur  simultane- 
ously, very  often  the  traveling  wave  precedes  the  stationary 
wave. 

The  phenomenon  may  start  with  a  traveling  wave  or  impulse, 
and  this,  by  reflection  at  the  ends  of  the  circuit,  and  combination 
of  the  reflected  waves  and  the  main  waves,  gradually  changes  to  a 
stationary  wave.  In  this  case,  the  traveling  wave  has  the  same 
frequency  as  the  stationary  wave  resulting  from  it.  In  Fig.  47  is 
shown  the  reproduction  of  an  oscillogram  of  the  formation  of  a 
stationary  oscillation  in  a  transmission  line  by  the  repeated  re- 


i, 


Fig.  47. —  CD11168.  —  Reproduction  of  an  Oscillogram  of  Stationary  Line 
Oscillation  by  Reflection  of  Impulse  from  Ends  of  Line. 

flection  from  the  ends  of  the  line  of  the  single  impulse  caused  by 
short  circuiting  the  energized  line  at  one  end.  In  the  beginning  of 
a  stationary  oscillation  of  a  compound  circuit,  that  is,  a  circuit  com- 
prising sections  of  different  constants,  traveling  waves  frequently 
occur,  by  which  the  energy  stored  magnetically  or  dielectrically  in 
the  different  circuit  sections  adjusts  itself  to  the  proportion  cor- 
responding to  the  stationary  oscillation  of  the  complete  circuit. 
Such  traveling  waves  then  are  local,  and  therefore  of  much  higher 
frequency  than  the  final  oscillation  of  the  complete  circuit,  and 
thus  die  out  at  a  faster  rate.  Occasionally  they  are  shown  by  the 
oscillograph  as  high-frequency  oscillations  intervening  between 


102      ELECTRIC  DISCHARGES,   WAVES  AND  IMPULSES. 

the  alternating  waves  before  the  beginning  of  the  transient  and 
the  low-frequency  stationary  oscillation  of  the  complete  circuit. 
Such  oscillograms  are  given  in  Figs.  48  to  49. 

Fig.  48A  gives  the  oscillation  of  the  compound  circuit  consisting 
of  28  miles  of  line  and  the  high-tension  winding  of  the  2500-kw. 
step-up  transformer,  caused  by  switching  off,  by  low-tension 
switches,  from  a  substation  at  the  end  of  a  153-mile  three-phase 
transmission  line,  at  88  kilovolts. 


Fig.  4SA. — CD10002.  —  Oscillogram  of  High-frequency  Oscillation  Preced- 
ing Low-frequency  Oscillation  of  Compound  Circuit  of  28  Miles  of 
100,000-volt  Line  and  Step-up  Transformer;  Low-tension  Switching. 

Fig.  48#  gives  the  oscillation  of  the  compound  circuit  consisting 
of  154  miles  of  three-phase  line  and  10,000-kw.  step-down  trans- 
former, when  switching  this  line,  by  high-tension  switches,  off  the 
end  of  another  154  miles  of  three-phase  line,  at  107  kilovolts. 
The  voltage  at  the  end  of  the  supply  line  is  given  as  ei,  at  the 
beginning  of  the  oscillating  circuit  as  e2. 

Fig.  49  shows  the  oscillations  and  traveling  waves  appearing 
in  a  compound  circuit  consisting  of  154  miles  of  three-phase  line 
and  10,000-kw.  step-down  transformer,  by  switching  it  on  and 
off  the  generating  system,  by  high-tension  switches,  at  89  kilo- 
volts. 

Frequently  traveling  waves  are  of  such   high   frequency  — 
reaching  into  the  millions  of  cycles  —  that  the  oscillograph  does 
not  record  them,  and  their  existence  and  approximate  magnitude 
are  determined  by  inserting  a  very  small  inductance  into  the 


TRAVELING  WAVES. 


103 


104      ELECTRIC  DISCHARGES,   WAVES  AND  IMPULSES. 

circuit  and  measuring  the  voltage  across  the  inductance  by  spark 
gap.  These  traveling  waves  of  very  high  frequency  are  extremely 
local,  often  extending  over  a  few  hundred  feet  only. 

An  approximate  estimate  of  the  effective  frequency  of  these  very 
high  frequency  local  traveling  waves  can  often  be  made  from  their 
striking  distance_across  a  small  inductance,  by  means  of  the 

relation  -^  =  V/  7^  =  z0,  discussed  in  Lecture  VI. 
lo        *  Co 

For  instance,  in  the  100,000- volt  transmission  line  of  Fig.  48A, 
the  closing  of  the  high-tension  oil  switch  produces  a  high-frequency 
oscillation  which  at  a  point  near  its  origin,  that  is,  near  the  switch, 
jumps  a  spark  gap  of  3.3  cm.  length,  corresponding  to  ei  =  35,000 
volts,  across  the  terminals  of  a  small  inductance  consisting  of  34 
turns  of  1.3  cm.  copper  rod,  of  15  cm.  mean  diameter  and  80  cm. 
length.  The  inductance  of  this  coil  is  calculated  as  approximately 
13  microhenrys.  The  line  constants  are,  L  =  0.323  henry,  C  = 

2.2  X  10~6  farad;  hence  z0  =  y  5  =  Vo.1465  X  103  =  383  ohms. 

The  sudden  change  of  voltage  at  the  line  terminals,  produced 

i  on  nno 
by  closing  the  switch,   is  -   -~—  =  57,700  volts  effective,  or  a 

V_3 

maximum  of  e0  =  57,700  X  V2  =  81,500  volts,  and  thus  gives 
a  maximum  transient  current  in  the  impulse,  of  i0  =  —  =  212 

amperes.  iQ  =  212  amperes  maximum,  traversing  the  inductance 
of  13  microhenrys,  thus  give  the  voltage,  recorded  by  the  spark 
gap,  of  e\  =  35,000.  If  then  /  =  frequency  of  impulse,  it  is 

e\  =  2-jrfLiQ. 

Or'  '=2^'          ;  .'  .        Y 

35,000 


27rX  13  X  10-6  X212 
=  2,000,000  cycles. 

37.  A  common  form  of  traveling  wave  is  the  discharge  of  a 
local  accumulation  of  stored  energy,  as  produced  for  instance  by 
a  direct  or  induced  lightning  stroke,  or  by  the  local  disturbance 
caused  by  a  change  of  circuit  conditions,  as  by  switching,  the 
blowing  of  fuses,  etc. 


TRAVELING  WAVES.  105 

Such  simple  traveling  waves  frequently  are  called  "impulses." 
When  such  an  impulse  passes  along  the  line,  at  any  point  of 
the  line,  the  wave  energy  is  zero  up  to  the  moment  where  the 
wave  front  of  the  impulse  arrives.  The  energy  then  rises,  more 
or  less  rapidly,  depending  on  the  steepness  of  the  wave  front, 
reaches  a  maximum,  and  then  decreases  again,  about  as  shown  in 
Fig.  50.  The  impulse  thus  is  the  combination  of  two  waves, 


Fig.  50.  —  Traveling  Wave. 


one,  which  decreases  very  rapidly,  e~(u  +  s}i}  and  thus  preponder- 
ates in  the  beginning  of  the  phenomenon,  and  one,  which  decreases 
slowly,  e-(u~s)t.  Hence  it  may  be  expressed  in  the  form: 

a2e-2^-s)^e-2sX,  (20) 


where  the  value  of  the  power-transfer  constant  s  determines  the 
"  steepness  of  wave  front." 

Figs.  51  to  53  show  oscillograms  of  the  propagation  of  such  an 
impulse  over  an  (artificial)  transmission  line  of  130  miles,*  of  the 
constants  : 

r  =  93.6  ohms, 

L  =  0.3944  henrys, 

C  =  1.135  microfarads,— 

thus  of  surge  impedance  ZQ  =  y  ~  =  590  ohms. 

The  impulse  is  produced  by  a  transformer  charge,  f 

Its  duration,  as  measured  from  the  oscillograms,  is  TQ  =  0.0036 

second. 

In  Fig.  51,  the  end  of  the  transmission  line  was  connected  to  a 

noninductive  resistance  equal  to  the  surge  impedance,  so  as  to 

*  For  description  of  the  line  see  "Design,  Construction,  and  Test  of  an  Arti- 
ficial Transmission  Line,"  by  J.  H.  Cunningham,  Proceedings  A.I.E.E.,  January, 
1911. 

t  In  the  manner  as  described  in  "Disruptive  Strength  of  Air  and  Oil  with 
Transient  Voltages,"  by  J.  L.  R.  Hayden  and  C.  P.  Steinmetz,  Transactions 
A.I.E.E.,  1910,  page  1125.  The  magnetic  energy  of  the  transformer  is,  however, 
larger,  about  4  joules,  and  the  transformer  contained  an  air  gap,  to  give  constant 
inductance. 


106      ELECTRIC  DISCHARGES,    WAVES  AND   IMPULSES. 


Fig.    51.  — CD11145.  —  Reproduction    of    Oscillogram    of    Propagation    of 
Impulse  Over  Transmission  Line;  no  Reflection.     Voltage, 


Fig.  52. —  CD  11 152.  —  Reproduction  of  Oscillogram  of  Propagation  of  Im- 
pulse Over  Transmission  Line; Reflection  from  Open  End  of  Line.     Voltage. 


TRAVELING  WAVES. 


107 


give  no  reflection.  The  upper  curve  shows  the  voltage  of  the 
impulse  at  the  beginning,  the  middle  curve  in  the  middle,  and  the 
lower  curve  at  the  end  of  the  line. 

Fig.  52  gives  the  same  three  voltages,  with  the  line  open  at  the 
end.  This  oscillogram  shows  the  repeated  reflections  of  the  vol- 
tage impulse  from  the  ends  of  the  line, — the  open  end  and  the 
transformer  inductance  at  the  beginning.  It  also  shows  the  in- 
crease of  voltage  by  reflection. 


Fig.  53. — CD11153.  —  Reproduction  of  Oscillogram  of  Propagation  of  Im- 
pulse Over  Transmission  Line;  Reflection  from  Open  End  of  Line. 
Current. 


Fig.  53  gives  the  current  impulses  at  the  beginning  and  the  mid- 
dle of  the  line,  corresponding  to  the  voltage  impulses  in  Fig.  52. 
This  oscillogram  shows  the  reversals  of  current  by  reflection,  and 
the  formation  of  a  stationary  oscillation  by  the  successive  reflec- 
tions of  the  traveling  wave  from  the  ends  of  the  line. 


LECTURE   IX. 
OSCILLATIONS  OF  THE   COMPOUND   CIRCUIT. 

38.  The  most  interesting  and  most  important  application  of 
the  traveling  wave  is  that  of  the  stationary  oscillation  of  a  com- 
pound circuit,  as  industrial  circuits  are  never  uniform,  but  consist 
of  sections  of  different  characteristics,  as  the  generating  system, 
transformer,  line,  load,  etc.  Oscillograms  of  such  circuits  have 
been  shown  in  the  previous  lecture. 

If  we  have  a  circuit  consisting  of  sections  1,  2,  3  .  .  .  ,  of  the 
respective  lengths  (in  velocity  measure)  Xi,  X2,  X3  .  .  .  ,  this 
entire  circuit,  when  left  to  itself,  gradually  dissipates  its  stored 
energy  by  a  transient.  As  function  of  the  time,  this  transient 
must  decrease  at  the  same  rate  u0  throughout  the  entire  circuit. 
Thus  the  time  decrement  of  all  the  sections  must  be 

6-**. 

Every  section,  however,  has  a  power-dissipation  constant,  u\t  Uz, 
u3  .  .  .  ,  which  represents  the  rate  at  which  the  stored  energy 
of  the  section  would  be  dissipated  by  the  losses  of  power  in  the 
section, 

€-"»',   €-«*',   €-"*'    .     .     . 

But  since  as  part  of  the  whole  circuit  each  section  must  die 
down  at  the  same  rate  e~Uot,  in  addition  to  its  power-dissipation 
decrement  e~Ul*,  e~"2'  .  .  .  ,  each  section  must  still  have  a  second 
time  decrement,  €-(«*-*J*,  e-(u0-u,)t  t  t  t  This  latter  does  not 
represent  power  dissipation,  and  thus  represents  power  transfer. 
That  is, 

51  =  U0  —  Ui, 

52  =  UQ  —  Uz,  (1) 


It  thus  follows  that  in  a  compound  circuit,  if  u0  is  the  average 
exponential  time  decrement  of  the  complete  circuit,  or  the  average 

108 


OSCILLATIONS  OF  THE  COMPOUND  CIRCUIT.  109 

power-dissipation  constant  of  the  circuit,  and  u  that  of  any  section, 
this  section  must  have  a  second  exponential  time  decrement, 

S  =  UQ  —  U,  (2) 

which  represents  power  transfer  from  the  section  to  other  sections, 
or,  if  s  is  negative,  power  received  from  other  sections.  The  oscil- 
lation of  every  individual  section  thus  is  a  traveling  wave,  with  a 
power-transfer  constant  s. 

As  UQ  is  the  average  dissipation  constant,  that  is,  an  average  of 
the  power-dissipation  constants  u  of  all  the  sections,  and  s  =  UQ  —  u 
the  power-transfer  constant,  some  of  the  s  must  be  positive,  some 
negative. 

In  any  section  in  which  the  power-dissipation  constant  u  is  less 
than  the  average  UQ  of  the  entire  circuit,  the  power-transfer  con- 
stant s  is  positive ;  that  is,  the  wave,  passing  over  this  section,  in- 
creases in  intensity,  builds  up,  or  in  other  words,  gathers  energy, 
which  it  carries  away  from  this  section  into  other  sections.  In 
any  section  in  which  the  power- dissipation  constant  u  is  greater 
than  the  average  UQ  of  the  entire  circuit,  the  power-transfer  con- 
stant s  is  negative;  that  is,  the  wave,  passing  over  this  section, 
decreases  in  intensity  and  thus  in  energy,  or  in  other  words,  leaves 
some  of  its  energy  in  this  section,  that  is,  supplies  energy  to  the 
section,  which  energy  it  brought  from  the  other  sections. 

By  the  power-transfer  constant  s,  sections  of  low  energy  dissi- 
pation supply  power  to  sections  of  high  energy  dissipation. 

39.  Let  for  instance  in  Fig.  43  be  represented  a  circuit  consist- 
ing of  step-up  transformer,  transmission  line,  and  load.  (The 
load,  consisting  of  step-down  transformer  and  its  secondary  cir- 
cuit, may  for  convenience  be  considered  as  one  circuit  section.) 
Assume  now  that  the  circuit  is  disconnected  from  the  power  sup- 
ply by  low-tension  switches,  at  A.  This  leaves  transformer,  line, 
and  load  as  a  compound  oscillating  circuit,  consisting  of  four 
sections:  the  high-tension  coil  of  the  step-up  transformer,  the  two 
lines,  and  the  load. 

Let  then  Xi  =  length  of  line,  X2  =  length  of  transformer  circuit, 
and  Xs  =  length  of  load  circuit,  in  velocity  measure.*  If  then 

*  If  Zi  =  length  of  circuit  section  in  any  measure,  and  L0  =  inductance, 
Co  =  capacity  per  unit  of  length  Zi,  then  the  length  of  the  circuit  in  velocity 
measure  is  Xi  =  o-oZi,  where  <TO  =  v  L0Co. 

Thus,  if  L  =  inductance,  C  =  capacity  per  transformer  coil,  n  =  number  of 
transformer  coils,  for  the  transformer  the  unit  of  length  is  the  coil;  hence  the 


110      ELECTRIC  DISCHARGES,   WAVES  AND  IMPULSES. 

HI  =  900  =  power-  dissipation  constant  of  the  line,  u*  =  100  = 
power-dissipation  constant  of  transformer,  and  uz  =  1600  =  power- 
dissipation  constant  of  the  load,  and  the  respective  lengths  of  the 
circuit  sections  are 

Xi  =  1.5  X  10-3;    X2  =  1  X  10~3;    \3  =  0.5  X  10~3, 
it  is: 

Line.         Transformer.         Line.  Load.  Sum. 

Length:  X  =  1.5X10-3  1X10~3  1.5X10~3  .5X10~3  4.5X1Q-3 
Power-dissipa- 

tion constant:      u  =         900  100            900  1600 

«X  =       1.35  .1  1.35  .8                3.6 

hence,  Wo=  average,  u  =  —  —  =  800,  and: 

ZA 

Power-transfer 
constant:    S  =  MO-M=      -100  +700        -100  -800 

The  transformer  thus  dissipates  power  at  the  rate  u2  =  100, 
while  it  sends  out  power  into  the  other  sections  at  the  rate  of 
s2  =  700,  or  seven  times  as  much  as  it  dissipates.  That  is,  it  sup- 
plies seven-eighths  of  its  stored  energy  to  other  sections.  The  load 
dissipates  power  at  the  rate  Uz  =  1600,  and  receives  power  at  the 
rate  —s  =  800;  that  is,  half  of  the  power  which  it  dissipates  is 
supplied  from  the  other  sections,  in  this  case  the  transformer. 

The  transmission  line  dissipates  power  at  the  rate  HI  =  900, 
that  is,  only  a  little  faster  than  the  average  power  dissipation  of 
the  entire  circuit,  u0  =  800;  and  the  line  thus  receives  power  only 
at  the  rate  —s=  100,  that  is,  receives  only  one-ninth  of  its  power 
from  the  transformer;  the  other  eight-ninths  come  from  its  stored 
energy. 

The  traveling  wave  passing  along  the  circuit  section  thus 
increases  or  decreases  in  its  power  at  the  rate  e+2*x;  that  is, 
in  the  line: 

p  =  pie~200X,  the   energy   of   the   wave    decreases   slowly; 
in  the  transformer: 

p  =  7?2C+1400X,  the  energy   of  the  wave   increases   rapidly; 


length  li  =  n,  and  the  length  in  velocity  measure,  X  =  aQn  =  n  VLC.  Or,  if 
L  =  inductance,  C  =  capacity  of  the  entire  transformer,  its  length  in  velocity 
measure  is  X  =  v  LC. 

Thus,  the  reduction  to  velocity  measure  of  distance  is  very  simple. 


OSCILLATIONS  OF  THE  COMPOUND  CIRCUIT, 


111 


in  the  load: 


p  —  p3e~l600X,  the  energy  of  the  wave  decreases  rapidly. 

Here  the  coefficients  of  pi,  p2,  PS  must  be  such  that  the  wave  at  the 
beginning  of  one  section  has  the  same  value  as  at  the  end  of  the 
preceding  section. 

In  general,  two  traveling  waves  run  around  the  circuit  in 
opposite  direction. 

Each  of  the  two  waves  reaches  its  maximum  intensity  in  this 
circuit  at  the  point  where  it  leaves  the  transformer  and  enters 
the  line,  since  in  the  transformer  it  increases,  while  in  the  line  it 
again  decreases,  in  intensity. 


Fig.  54.  —  Energy  Distribution  in  Compound  Oscillation  of  Closed  Circuit; 

High  Line  Loss. 

Assuming  that  the  maximum  value  of  the  one  wave  is  6,  that  of 
the  opposite  wave  4  megawatts,  the  power  values  of  the  two  waves 
then  are  plotted  in  the  upper  part  of  Fig.  54,  and  their  difference, 
that  is,  the  resultant  flow  of  power,  in  the  lower  part  of  Fig.  54. 
As  seen  from  the  latter,  there  are  two  power  nodes,  or  points  over 
which  no  power  flows,  one  in  the  transformer  and  one  in  the  load, 
and  the  power  flows  from  the  transformer  over  the  line  into  the 
load;  the  transformer  acts  as  generator  of  the  power,  and  of  this 


112      ELECTRIC  DISCHARGES,   WAVES  AND  IMPULSES. 

power  a  fraction  is  consumed  in  the  line,  the  rest  supplied  to  the 
load. 

40.  The  diagram  of  this  transient  power  transfer  of  the  system 
thus  is  very  similar  to  that  of  the  permanent  power  transmis- 
sion by  alternating  currents:  a  source  of  power,  a  partial  con- 
sumption in  the  line,  and  the  rest  of  the  power  consumed  in 
the  load. 

However,  this  transient  power-transfer  diagram  does  not  repre- 
sent the  entire  power  which  is  being  consumed  in  the  circuit,  as 
power  is  also  supplied  from  the  stored  energy  of  the  circuit;  and 
the  case  may  thus  arise  —  which  cannot  exist  in  a  permanent 
power  transmission  —  that  the  power  dissipation  of  the  line  is 
less  than  corresponds  to  its  stored  energy,  and  the  line  also  supplies 
power  to  the  load,  that  is,  acts  as  generator,  and  in  this  case  the 
power  would  not  be  a  maximum  at  the  transformer  terminals, 
but  would  still  further  increase  in  the  line,  reaching  its  maxi- 
mum at  the  load  terminals.  This  obviously  is  possible  only 
with  transient  power,  where  the  line  has  a  store  of  energy  from 
which  it  can  draw  in  supplying  power.  In  permanent  condition 
the  line  could  not  add  to  the  power,  but  must  consume,  that  is, 
the  permanent  power-transmission  diagram  must  always  be  like 
Fig.  54. 

Not  so,  as  seen,  with  the  transient  of  the  stationary  oscillation. 
Assume,  for  instance,  that  we  reduce  the  power  dissipation  in 
the  line  by  doubling  the  conductor  section,  that  is,  reducing  the 
resistance  to  one-half.  As  L  thereby  also  slightly  decreases, 
C  increases,  and  g  possibly  changes,  the  change  brought  about  in 

the  constant  u  =  =lj  ~^7>)  *s  no^  necessarily  a  reduction  to 

half,  but  depends  upon  the  dimensions  of  the  line.  Assuming 
therefore,  that  the  power-dissipation  constant  of  the  line  is  by  the 
doubling  of  the  line  section  reduced  from  u\  =  900  to  HI  =  500, 
this  gives  the  constants: 

Line.  Transformer.         Line.  Load.  Sum. 

X=        1.5X10-3      1X10-3      1.5X10-3      .5X10-3    4.5X1Q-3 
u=  500  100  500  1600 

wX=  .75  .1  .75  .8  2.4 

SwX 
hence,  MO  =  average,  u  =  — -  =  533,  and: 

2/A 

s=  +33          +433  +33          -1067 


one- 


OSCILLATIONS  OF  THE  COMPOUND   CIRCUIT. 


113 


That  is,  the  power-transfer  constant  of  the  line  has  become  posi- 
tive, si  =  33,  and  the  line  now  assists  the  transformer  in  supplying 
power  to  the  load.  Assuming  again  the  values  of  the  two  travel- 
ing waves,  where  they  leave  the  transformer  (which  now  are  not 
the  maximum  values,  since  the  waves  still  further  increase  in 
intensity  in  passing  over  the  lines),  as  6  and  4  megawatts  respec- 
tively, the  power  diagram  of  the  two  waves,  and  the  power  dia- 
gram of  their  resultant,  are  given  in  Fig.  55. 


Fig.  55.  —  Energy  Distribution  in  Compound  Oscillation  of  Closed  Circuit; 

Low  Line  Loss. 

In  a  closed  circuit,  as  here  discussed,  the  relative  intensity  of 
the  two  component  waves  of  opposite  direction  is  not  definite, 
but  depends  on  the  circuit  condition  at  the  starting  moment  of 
the  transient. 

In  an  oscillation  of  an  open  compound  circuit,  the  relative 
intensities  of  the  two  component  waves  are  fixed  by  the  condition 
that  at  the  open  ends  of  the  circuit  the  power  transfer  must  be 
zero. 

As  illustration  may  be  considered  a  circuit  comprising  the  high- 
potential  coil  of  the  step-up  transformer,  and  the  two  lines,  which 
are  assumed  as  open  at  the  step-down  end,  as  illustrated  diagram- 
matically  in  Fig.  56. 


114      ELECTRIC  DISCHARGES,   WAVES  AND  IMPULSES. 


Choosing   the   same   lengths  and  the  same   power-dissipation 
constants  as  in  the  previous  illustrations,  this  gives: 


Line.  Transformer.  Line.                  Sum. 

1.5X10-3  1X10-3  1.5X10-3         4X10-3 

900  100  900 

1.35  .1  1.35                  2.8 

SwX 


x= 

u\  = 
hence,  w0  =  average,  u  =  ^^  =  700,  and: 

s=  -200  +600  -200 

Line 


Transformer 


Line 


Fig.  56. 

The  diagram  of  the  power  of  the  two  waves  of  opposite  direc- 
tions, and  of  the  resultant  power,  is  shown  in  Fig.  57,  assuming 
6  megawatts  as  the  maximum  power  of  each  wave,  which  is  reached 
at  the  point  where  it  leaves  the  transformer. 


Transmission  Line      Transformer    Transmission  Line 


U  =900 


U  =100 


U=900 


Fig.  57.  —  Energy  Distribution  in  Compound  Oscillation  of  Open  Circuit. 

In  this  case  the  two  waves  must  be  of  the  same  intensity,  so 
as  to  give  0  as  resultant  at  the  open  ends  of  the  line.  A  power 
node  then  appears  in  the  center  of  the  transformer. 

41.  A  stationary  oscillation  of  a  compound  circuit  consists  of 
two  traveling  waves,  traversing  the  circuit  in  opposite  direction, 
and  transferring  power  between  the  circuit  section  in  such  a  manner 


OSCILLATIONS  OF  THE  COMPOUND  CIRCUIT.  115 

as  to  give  the  same  rate  of  energy  dissipation  in  all  circuit  sections. 
As  the  result  of  this  power  transfer,  the  stored  energy  of  the 
system  must  be  uniformly  distributed  throughout  the  entire 
circuit,  and  if  it  is  not  so  in  the  beginning  of  the  transient,  local 
traveling  waves  redistribute  the  energy  throughout  the  oscillat- 
ing circuit,  as  stated  before.  Such  local  oscillations  are  usually 
of  very  high  frequency,  but  sometimes  come  within  the  range  of 
the  oscillograph,  as  in  Fig.  47. 

During  the  oscillation  of  the  complex  circuit,  every  circuit 
element  d\  (in  velocity  measure),  or  every  wave  length  or  equal 
part  of  the  wave  length,  therefore  contains  the  same  amount  of 
stored  energy.  That  is,  if  e0  =  maximum  voltage,  i0  =  maximum 

current,  and  X0  =  wave  length,  the  average  energy    °  °  Q  must 

be  constant  throughout  the  entire  circuit.  Since,  however,  in 
velocity  measure,  Xo  is  constant  and  equal  to  the  period  TO  through- 
out all  the  sections  of  the  circuit,  the  product  of  maximum  voltage 
and  of  maximum  current,  e0^o,  thus  must  be  constant  throughout 
the  entire  circuit. 

The  same  applies  to  an  ordinary  traveling  wave  or  impulse. 
Since  it  is  the  same  energy  which  moves  along  the  circuit  at  a 
constant  rate,  the  energy  contents  for  equal  sections  of  the  circuit 
must  be  the  same  except  for  the  factor  e~ 2  "*,  by  which  the  energy 
decreases  with  the  time,  and  thus  with  the  distance  traversed 
during  this  time. 

Maximum  voltage  e0  and  maximum  current  i'o,  however,  are 
related  to  each  other  by  the  condition_ 

e°  i  /^°  fo\ 

—  =  ZQ  =  y  -FT  ,  (3) 

and  as  the  relation  of  L0  and  <70  is  different  in  the  different  sections, 
and  that  very  much  so,  ZQ,  and  with  it  the  ratio  of  maximum 
voltage  to  maximum  current,  differ  for  the  different  sections  of 
the  circuit. 

If  then  ei  and  ii  are  maximum  voltage  and  maximum  current 

respectively  of  one  section,  and  z\  =  y  -^  is  the  "natural  imped- 
ance "  of  this  section,  and  ez,  12,  and  z2  —  V/TT  are  the  correspond- 

V    02 

ing  values  for  another  section,  it  is 

•          _  •  / '  A\ 


116      ELECTRIC  DISCHARGES,   WAVES  AND  IMPULSES. 

and  since 

^  —        ei  — 
iz         '    ii 

substituting  e2  =  i'2z2,l 

=  'z  I  ^ 

into  (4)  gives 

or 


and 

z2 
or 

fz» 


/ON 

That  is,  in  the  same  oscillating  circuit,  the  maximum  voltages 
60  in  the  different  sections  are  proportional  to,  and  the  maximum 
currents  i0  inversely  proportional  to,  the  square  root  of  the  natural 
impedances  z0  of  the  sections,  that  is,  to  the  fourth  root  of  the 

ratios  of  inductance  to  capacity  -^  • 

to 

At  every  transition  point  between  successive  sections  traversed 
by  a  traveling  wave,  as  those  of  an  oscillating  system,  a  trans- 
formation of  voltage  and  of  current  occurs,  by  a  transformation 
ratio  which  is  the  square  root  of  the  ratio  of  the  natural  imped- 
ances, ZQ  =  V  TT  >  of  the  two  respective  sections. 
*  Co 

When  passing  from  a  section  of  high  capacity  and  low  induc- 
tance, that  is,  low  impedance  z0,  to  a  section  of  low  capacity  and 
high  inductance,  that  is,  high  impedance  z0,  as  when  passing  from 
a  transmission  line  into  a  transformer,  or  from  a  cable  into  a  trans- 
mission line,  the  voltage  thus  is  transformed  up,  and  the  current 
transformed  down,  and  inversely,  with  a  wave  passing  in  opposite 
direction. 

A  low-voltage  high-current  wave  in  a  transmission  line  thus 
becomes  a  high-voltage  low-current  wave  in  a  transformer,  and 
inversely,  and  thus,  while  it  may  be  harmless  in  the  line,  may 
become  destructive  in  the  transformer,  etc. 


OSCILLATIONS  OF  THE  COMPOUND  CIRCUIT. 


117 


42.  At  the  transition  point  between  two  successive  sections, 
the  current  and  voltage  respectively  must  be  the  same  in  the 
two  sections.  Since  the  maximum  values  of  current  and  voltage 
respectively  are  different  in  the  two  sections,  the  phase  angles  of 
the  waves  of  the  two  sections  must  be  different  at  the  transition 
point;  that  is,  a  change  of  phase  angle  occurs  at  the  transition 
point. 

This  is  illustrated  in  Fig.  58.  Let  z0  =  200  in  the  first  section 
(transmission  line),  ZQ  =  800  in  the  second  section  (transformer). 

/800 

The  transformation  ratio  between  the  sections  then  is  V  onn=2; 

»   ^Uu 

that  is,  the  maximum  voltage  of  the  second  section  is  twice,  and 
the  maximum  current  half,  that  of  the  first  section,  and  the  waves 
of  current  and  of  voltage  in  the  two  sections  thus  may  be  as 
illustrated  for  the  voltage  in  Fig.  58,  by  e\e^. 


Fig.  58.  —  Effect  of  Transition  Point  on  Traveling  Wave. 


If  then  ef  and  if  are  the  values  of  voltage  and  current  respec- 
tively at  the  transition  point  between  two  sections  1  and  2,  and 
e\  and  ii  the  maximum  voltage  and  maximum  current  respec- 
tively of  the  first,  e%  and  iz  of  the  second,  section,  the  voltage  phase 
and  current  phase  at  the  transition  point  are,  respectively: 


For  the  wave  of  the  first  section: 

_  =  cos  71  and    -r  =  cos  5i. 

For  the  wave  of  the  second  section: 

e'  i' 

—  —  cos  72  and    —  =  cos  §2. 


(9) 


118      ELECTRIC  DISCHARGES,   WAVES  AND  IMPULSES. 
Dividing  the  two  pairs  of  equations  of  (9)  gives 

cos  72  _  61 
cos  71      62 


=  ii  =  Jz*  f 
iz      V  ?! 


cos  <5i 
hence,  multiplied, 

cos  72      cos  82 

cos  71      cos  di 
or  (11) 

cos  72  _  cos  di 

COS  7i         COS  £2 

or 

cos  71  cos  5i  =  cos  72  cos  52; 

that  is,  the  ratio  of  the  cosines  of  the  current  phases  at  the  tran- 
sition point  is  the  reciprocal  of  the  ratio  of  the  cosines  of  the 
voltage  phases  at  this  point. 

Since  at  the  transition  point  between  two  sections  the  voltage 
and  current  change,  from  ei,  ii  to  62,  is,  by  the  transformation  ratio 

— ,  this  change  can  also  be  represented  as  a  partial  reflection. 

That  is,  the  current  i\  can  be  considered  as  consisting  of  a  compo- 
nent z'2,  which  passes  over  the  transition  point,  is  "  transmitted  " 
current,  and  a  component  i\  =  i\  —  iz,  which  is  "  reflected  " 
current,  etc.  The  greater  then  the  change  of  circuit  constants 
at  the  transition  point,  the  greater  is  the  difference  between  the 
currents  and  voltages  of  the  two  sections;  that  is,  the  more  of 
current  and  voltage  are  reflected,  the  less  transmitted,  and  if  the 
change  of  constants  is  very  great,  as  when  entering  from  a  trans- 
mission line  a  reactance  of  very  low  capacity,  almost  all  the 
current  is  reflected,  and  very  little  passes  into  and  through  the 
reactance,  but  a  high  voltage  is  produced  in  the  reactance. 


v/ 


LECTURE  X. 

INDUCTANCE  AND   CAPACITY   OF  ROUND 
PARALLEL   CONDUCTORS. 

A.   Inductance  and  capacity. 

43.  As  inductance  and  capacity  are  the  two  circuit  constants 
which  represent  the  energy  storage,  and  which  therefore  are  of 
fundamental  importance  in  the  study  of  transients,  their  calcula- 
tion is  discussed  in  the  following. 

The  inductance  is  the  ratio  of  the  interlinkages  of  the  mag- 
netic flux  to  the  current, 

£  =  ?-  (i) 

i/ 

where  <i>  =  magnetic  flux  or  number  of  lines  of  magnetic  force, 
and  n  the  number  of  times  which  each  line  of  magnetic  force 
interlinks  with  the  current  i. 

The  capacity  is  the  ratio  of  the  dielectric  flux  to  the  voltage, 


where  \f/  is  the  dielectric  flux,  or  number  of  lines  of  dielectric 
force,  and  e  the  voltage  which  produces  it. 

With  a  single  round  conductor  without  return  conductor  (as 
wireless  antennae)  or  with  the  return  conductor  at  infinite  dis- 
tance, the  lines  of  magnetic  force  are  concentric  circles,  shown  by 
drawn  lines  in  Fig.  8,  page  10,  and  the  lines  of  dielectric  force 
are  straight  lines  radiating  from  the  conductor,  shown  dotted  in 
Fig.  8. 

Due  to  the  return  conductor,  in  a  two-wire  circuit,  the  lines  of 
magnetic  and  dielectric  force  are  crowded  together  between  the 
conductors,  and  the  former  become  eccentric  circles,  the  latter 
circles  intersecting  in  two  points  (the  foci)  inside  of  the  con- 
ductors, as  shown  in  Fig.  9,  page  11.  With  more  than  one  return 
conductor,  and  with  phase  displacement  between  the  return 
currents,  as  in  a  three-phase  three-wire  circuit,  the  path  of  the 

119 


'iJBLtiGTRIC  DISCHARGES,   WAVES  AND  IMPULSES. 


lines  of  force  is  still  more  complicated,  and  varies  during  the 
cyclic  change  of  current. 

The  calculation  of  such  more  complex  magnetic  and  dielectric 
fields  becomes  simple,  however,  by  the  method  of  superposition  of 
fields.  As  long  as  the  magnetic  and  the  dielectric  flux  are  pro- 
portional respectively  to  the  current  and  the  voltage,  —  which  is 
the  case  with  the  former  in  nonmagnetic  materials,  with  the  latter 
for  all  densities  below  the  dielectric  strength  of  the  material,— 
the  resultant  field  of  any  number  of  conductors  at  any  point  in 
space  is  the  combination  of  the  component  fields  of  the  individual 
conductors. 


Fig.  59.  —  Magnetic  Field  of  Circuit. 

Thus  the  field  of  conductor  A  and  return  conductor  B  is  the 
combination  of  the  field  of  A,  of  the  shape  Fig.  8,  and  the  field  of 
B,  of  the  same  shape,  but  in  opposite  direction,  as  shown  for  the 
magnetic  fields  in  Fig.  59. 

All  the  lines  of  magnetic  force  of  the  resultant  magnetic  field 
must  pass  between  the  two  conductors,  since  a  line  of  magnetic 
force,  which  surrounds  both  conductors,  would  have  no  m.m.f., 
and  thus  could  not  exist.  That  is,  the  lines  of  magnetic  force  of 
A  beyond  B,  and  those  of  B  beyond  A,  shown  dotted  in  Fig.  59, 
neutralize  each  other  and  thereby  vanish;  thus,  in  determining 
the  resultant  magnetic  flux  of  conductor  and  return  conductor 
(whether  the  latter  is  a  single  conductor,  or  divided  into  two  con- 


ROUND  PARALLEL  CONDUCTORS. 


121 


ductors  out  of  phase  with  each  other,  as  in  a  three-phase  circuit), 
only  the  lines  of  magnetic  force  within  the  space  from  conductor 
to  return  conductor  need  to  be  considered.  Thus,  the  resultant 
magnetic  flux  of  a  circuit  consisting  of  conductor  A  and  return 
conductor  B,  at  distance  s  from  each  other,  consists  of  the  lines 
of  magnetic  force  surrounding  A  up  to  the  distance  s,  and  the 
lines  of  magnetic  force  surrounding  B  up  to  the  distance  s.  The 
former  is  attributed  to  the  inductance  of  conductor  A,  the  latter 
to  the  inductance  of  conductor  B.  If  both  conductors  have 
the  same  size,  they  give  equal  inductances;  if  of  unequal  size,  the 
smaller  conductor  has  the  higher  inductance.  In  the  same  manner 
in  a  three-phase  circuit,  the  inductance  of  each  of  the  three  con- 
ductors is  that  corresponding  to  the  lines  of  magnetic  force  sur- 
rounding the  respective  conductor,  up  to  the  distance  of  the  return 
conductor. 

B.   Calculation  of  inductance. 

44.  If  r  is  the  radius  of  the  conductor,  s  the  distance  of  the 
return  conductor,  in  Fig.  60,  the  magnetic  flux  consists  of  that 
external  to  the  conductor,  from  r  to  s,  and  that  internal  to  the 
conductor,  from  0  to  r. 


Fig.  60.  —  Inductance  Calculation  of  Circuit. 

At  distance  x  from  the  conductor  center,  the  length  of  the  mag 
netic  circuit  is  2  irx,  and  if  F  =  m.m.f.  of  the  conductor,  the  mag- 
netizing force  is 


and  the  field  intensity 

hence  the  magnetic  density 

(B 


2F 

x 


(4) 
(5) 


122      ELECTRIC  DISCHARGES,   WAVES  AND  IMPULSES. 
and  the  magnetic  flux  in  the  zone  dx  thus  is 

d^=^fdx,  I  (6) 

and  the  magnetic  flux  interlinked  with  the  conductor  thus  is 


X 

hence  the  total  magnetic  flux  between  the  distances  x\  and  z2  is 

rx*2 

thus  the  inductance 


X 


1.  External  magnetic  flux,     xi  =  r;  xz  =  s;  jP  =  i,  as  this  flux 
surrounds  the  total  current;  and  n  =  1,  as  each  line  of  magnetic 
force  surrounds  the  conductor  once,     ju  =  1  in  air,  thus: 

?-""-:-         <»> 

2.  Internal  magnetic  flux.    Assuming  uniform  current  ^density 
throughout  the  conductor  section,  it  is 


Cx\2 
-J  , 

as  each  line  of  magnetic  force  surrounds  only  a  part  of  the  con- 
ductor 


and  the  total  inductance  of  the  conductor  thus  is 

C        9       //  i 
L  =  LI  +  L2  =  2  j  log-  +T(  per  cm.  length  of  conductor,     (11) 

or,  if  the  conductor  consists  of  nonmagnetic  material,  ju  =  1  : 

(12) 


ROUND  PARALLEL  CONDUCTORS. 


123 


This  is  in  absolute  units,  and,  reduced  to  henry s,  =  109  absolute 
units : 


=  2  j  log  ?  + 1 1 10-9  h  per  cm. 


(13) 
(14) 


In  these  equations  the  logarithm  is  the  natural  logarithm,  which  is 
most  conveniently  derived  by  dividing  the  common  or  10  logarithm 
by  0.4343.* 

C.   Discussion  of  inductance. 

45.  In  equations  (11)  to  (14)  s  is  the  distance  between  the  con- 
ductors. If  s  is  large  compared  with  r,  it  is  immaterial  whether 
as  s  is  considered  the  distance  between  the  conductor  centers,  or 
between  the  insides,  or  outsides,  etc.;  and,  in  calculating  the  in- 
ductance of  transmission-line  conductors,  this  is  the  case,  and  it 
therefore  is  immaterial  which  distance  is  chosen  as  s;  and  usually, 
in  speaking  of  the  "distance  between  the  line  conductors,"  no 
attention  is  paid  to  the  meaning  of  s. 


Fig.  61.  —  Inductance  Calculation  of  Cable. 

However,  if  s  is  of  the  same  magnitude  as  r,  as  with  the  con- 
ductors of  cables,  the  meaning  of  s  has  to  be  specified. 

Let  then  in  Fig.  61  r  =  radius  of  conductors,  and  s  =  distance 
between  conductor  centers.  Assuming  uniform  current  density 
in  the  conductors,  the  flux  distribution  of  conductor  A  then  is  as 
indicated  diagrammatically  in  Fig.  61. 

*  0.4343  =  log10*, 


124      ELECTRIC  DISCHARGES,   WAVES  AND  IMPULSES. 

The  flux  then  consists  of  three  parts: 

3>i,  between  the  conductors,  giving  the  inductance 


and  shown  shaded  in  Fig.  61. 
$2,  inside  of  conductor  A,  giving  the  inductance 


$3,  the  flux  external  to  A,  which  passes  through  conductor  B 
and  thereby  incloses  the  conductor  A  and  part  of  the  conductor 

F 

J5,  and  thus  has  a  m.m.f.  less  than  i,  that  is,  gives  -  <  1. 

% 

That  is,  a  line  of  magnetic  force  at  distance  s  —  r<x<s  +  r 
incloses  the  part  q  of  the  conductor  B,  thus  incloses  the  fraction 

-y-  of  the  return  current,  and  thus  has  the  m.m.f. 

F  -1       q 

1  '     ~7v" 

An  exact  calculation  of  the  flux  <£3,  and  the  component  inductance 
LS  resulting  from  it,  is  complicated,  and,  due  to  the  nature  of  the 
phenomenon,  the  result  could  not  be  accurate;  and  an  approxima- 
tion is  sufficient  in  giving  an  accuracy  as  great  as  the  variability  of 
the  phenomenon  permits. 

The  magnetic  flux  $3  does  not  merely  give  an  inductance,  but, 
if  alternating,  produces  a  potential  difference  between  the  two 
sides  of  conductor  B,  and  thereby  a  higher  current  density  on  the 
side  of  B  toward  A;  and  as  this  effect  depends  on  the  conduc- 
tivity of  the  conductor  material,  and  on  the  frequency  of  the 
current,  it  cannot  be  determined  without  having  the  frequency, 
etc.,  given.  The  same  applies  for  the  flux  $1,  which  is  reduced  by 
unequal  current  density  due  to  its  screening  effect,  so  that  in  the 
limiting  case,  for  conductors  of  perfect  conductivity,  that  is,  zero 
resistance,  or  for  infinite,  that  is,  very  high  frequency,  only  the 
magnetic  flux  $1  exists,  which  is  shown  shaded  in  Fig.  5;  but  <J>2 
and  $3  are  zero,  and  the  inductance  is 

.  (15) 


ROUND  PARALLEL   CONDUCTORS. 


125 


That  is,  in  other  words,  with  small  conductors  and  moderate 
currents,  the  total  inductance  in  Fig.  61  is  so  small  compared 
with  the  inductances  in  the  other  parts  of  the  electric  circuit 
that  no  very  great  accuracy  of  its  calculation  is  required;  with 
large  conductors  and  large  currents,  however,  the  unequal  current 
distribution  and  resultant  increase  of  resistance  become  so  con- 
siderable, with  round  conductors,  as  to  make  their  use  uneconom- 
ical, and  leads  to  the  use  of  flat  conductors.  With  flat  conductors, 
however,  conductivity  and  frequency  enter  into  the  value  of  in- 
ductance as  determining  factors. 

The  exact  determination  of  the  inductance  of  round  parallel 
conductors  at  short  distances  from  each  other  thus  is  only  of 
theoretical,  but  rarely  of  practical,  importance. 

An  approximate  estimate  of  the  inductance  L3  is  given  by  con- 
sidering two  extreme  cases: 

(a)  The  return  conductor  is  of  the  shape  Fig.  62,  that  is,  from 
s  —  r  to  s  +  f  the  m.m.f.  varies  uniformly. 


B 


Fig.  62.  Fig.  63. 

Inductance  Calculation  of  Cable. 

(6)  The  return  conductor  is  of  the  shape  Fig.  63,  that  is,  the 
m.m.f.  of  the  return  conductor  increases  uniformly  from  s  —  r  to 
s,  and  then  decreases  again  from  s  to  s  +  r. 

(a)  For  s  —  r  <  x  <  s  +  r,  it  is 


-f  r  —  x 
2r 


2r        2r 


hence  by  (8), 


/»s+rg_|_r  fa        r*+*dx 
J8_r      r      x       Ja_r    r 


s  —  r 


by  the  approximation 

log  (1  ±  x)  = 


(16) 

(17) 
(18) 


126      ELECTRIC  DISCHARGES,   WAVES  AND  IMPULSES. 

it  is 

,      s  +  r      .      s  +  r      ,      s  —  r      ,      L    .  r\     ,     /.,       r\      rtr 
log—  =  log  —  -  log  —  =  log  (l  +  -)  -  log(l  -  -)  =  2-g, 

hence 


r 
(6)  For  s  —  r  <  x  <  s,  it  is 


f-l-sl^^r^h  (20) 


and  for  s  <  x  <  s  +  r,  it  is 

:'  '- 

hence, 


and  integrated  this  gives 

fc-aiog^  +  ^log'-f'-  ii^log^-3,  (23) 

o   —   if  o  /  o   —   / 

and  by  the  approximation  (18)  this  reduces  to 

L,-^,  (24) 

O 

that  is,  the  same  value  as  (19);  and  as  the  actual  case,  Fig.  60, 
should  lie  between  Figs.  61  and  62,  the  common  approximation  of 
the  latter  two  cases  should  be  a  close  approximation  of  case  4. 
That  is,  for  conductors  close  together  it  is 

L  =  L!  +  L2  +  L3 

(25) 


However,  -   can  be  considered  as  the  approximation  of  —  log 
s 

(  1  --  )=  log  -  ,  and  substituting  this  in  (25)  gives,  by  com- 
\        s/  s  —  r  _ 


o    _        y  o  o 

bining  log  --  h  log  -  =  log  -  : 

T  S         T  T 


(26) 


ROUND  PARALLEL  CONDUCTORS. 


127 


where  s  =  distance  between  conductor  centers,  as  the  closest 
approximation  in  the  case  where  the  distance  between  the  con- 
ductors is  small.  This  is  the  same  expression  as  (13). 

In  view  of  the  secondary  phenomena  unavoidable  in  the  con- 
ductors, equation  (26)  appears  sufficiently  accurate  for  all  practi- 
cal purposes,  except  when  taking  into  consideration  the  secondary 
phenomena,  as  unequal  current  distribution,  etc.,  in  which  case 
the  frequency,  conductivity,  etc.,  are  required. 

D.   Calculation  of  capacity. 

46.  The  lines  of  dielectric  force  of  the  conductor  A  are  straight 
radial  lines,  shown  dotted  in  Fig.  64,  and  the  dielectric  equipoten- 
tial  lines  are  concentric  circles,  shown  drawn  in  Fig.  64. 


Fig.  64.  —  Electric  Field  of  Conductor. 

If  e  =  voltage  between  conductor  A  and  return  conductor  B, 
and  s  the  distance  between  the  conductors,  the  potential  difference 
between  the  equipotential  line  at  the  surface  of  A,  and  the  equi- 
potential  line  which  traverses  B,  must  be  e. 

If  e  =  potential  difference  or  voltage,  and  I  =  distance,  over 
which  this  potential  difference  acts, 

G  =  -  =  potential  gradient,  or  electrifying  force,         (27) 


128      ELECTRIC  DISCHARGES,   WAVES  AND  IMPULSES. 

and  K  =  -  —  2  =  -  —  ^  =  dielectric  field  intensity,  (28) 

4  Trf        4  irV  L 

where  v2  is  the  reduction  factor  from  the  electrostatic  to  the 
electromagnetic  system  of  units,  and 

v  =  3  X  1010  cm.  sec.  =  velocity  of  light;  (29) 

the  dielectric  density  then  is 


where  K  =  specific  capacity  of  medium,  =  1  in  air.     The  dielectric 
flux  then  is 


where  A  =  section  of  dielectric  flux.     Or  inversely: 

-IS?*  :     ||         (32) 

If  then  ^  =  dielectric  flux,  in  Fig.  60,  at  a  distance  x  from  the 
conductor  A,  in  a  zone  of  thickness  dx,  and  section  2  TTZ,  the  voltage 

is,  by  (32), 

, 
de 


and  the  voltage  consumed  between  distances  x\  and  x2  thus  is 

/»*2          2v2^       Xz 
ei2  =    /     de  =  —^-Llog-,  (34) 

hence  the  capacity  of  this  space : 

C2 r K  /'Q^^ 

i   —  —5 —  *  (po) 


The  capacity  of  the  conductor  A  against  the  return  conductor  B 
then  is  the  capacity  of  the  space  from  the  distance  Zi  =  r  to  the 
distance  x^  =  s,  hence  is,  by  (35), 

C  = —  per  cm.  (36) 

2t;2log- 


ROUND  PARALLEL  CONDUCTORS.  129 

in  absolute  units,  hence,  reduced  to  farads, 

C=     *1Q9     /per  cm.,  (37) 

2z;2log- 

and  in  air,  for  K  =  1 : 

1H9 

(38) 


Immediately  it  follows:  the  external  inductance  was,  by  (9), 

Li  =  2  log-  10~9  h  per  cm., 
and  multiplying  this  with  (38)  gives 


or 


CL> = £' 


(39) 


that  is,  the  capacity  equals  the  reciprocal  of  the  external  inductance 
LI  times  the  velocity  square  of  light.  The  external  inductance  LI 
would  be  the  inductance  of  a  conductor  which  had  perfect  con- 
ductivity, or  zero  losses  of  power.  It  is 


VLC 

=  velocity  of  propagation  of  the  electric  field,  and  this  velocity  is 
less  than  the  velocity  of  light,  due  to  the  retardation  by  the  power 
dissipation  in  the  conductor,  and  becomes  equal  to  the  velocity  of 
light  v  if  there  is  no  power  dissipation,  and,  in  the  latter  case,  L 
would  be  equal  to  LI,  the  external  inductance. 

The  equation  (39)  is  the  most  convenient  to  calculate  capacities 
in  complex  systems  of  circuits  from  the  inductances,  or  inversely, 
to  determine  the  inductance  of  cables  from  the  measured  capacity, 
etc.  More  complete,  this  equation  is 

CLt  =  ^,  (40) 

where  K  =  specific  capacity  or  permittivity,  /*  =  permeability  of 
the  medium. 


130      ELECTRIC  DISCHARGES,   WAVES  AND  IMPULSES. 

E.   Conductor  with  ground  return. 

47.  As  seen  in  the  preceding,  in  the  electric  field  of  conductor  A 
and  return  conductor  B,  at  distance  s  from  each  other,  Fig.  9,  the 
lines  of  magnetic  force  from  conductor  A  to  the  center  line  CC' 
are  equal  in  number  and  in  magnetic  energy  to  the  lines  of  mag- 
netic force  which  surround  the  conductor  in  Fig.  59,  in  concentric 
circles  up  to  the  distance  s,  and  give  the  inductance  L  of  conductor 
A.  The  lines  of  dielectric  force  which  radiate  from  conductor  A 
up  to  the  center  line  CC',  Fig.  9,  are  equal  in  number  and  in  dielec- 
tric energy  to  the  lines  of  dielectric  force  which  issue  as  straight 
lines  from  the  conductor,  Fig.  8,  up  to  the  distance  s,  and  repre- 
sent the  capacity  C  of  the  conductor  A.  The  center  line  CC'  is  a 
dielectric  equipotential  line,  and  a  line  of  magnetic  force,  and  there- 
fore, if  it  were  replaced  by  a  conducting  plane  of  perfect  conduc- 
tivity, this  would  exert  no  effect  on  the  magnetic  or  the  dielectric 
field  between  the  conductors  A  and  B. 

If  then,  in  the  electric  field  between  overhead  conductor  and 
ground,  we  consider  the  ground  as  a  plane  of  perfect  conductivity, 
we  get  the  same  electric  field  as  between  conductor  A  and  central 
plane  CC'  in  Fig.  9.  That  is,  the  equations  of  inductance  and 
capacity  of  a^conductor  with  return  conductor  at  distance  s  can 
be  immediately  applied  to  the  inductance  and  capacity  of  a  con- 
ductor with  ground  return,  by  using  as  distance  s  twice  the  dis- 
tance of  the  conductor  from  the  ground  return.  That  is,  the 
inductance  and  capacity  of  a  conductor  with  ground  return  are 
the  same  as  the  inductance  and  capacity  of  the  conductor  against 
its  image  conductor,  that  is,  against  a  conductor  at  the  same  dis- 
tance below  the  ground  as  the  conductor  is  above  ground. 

As  the  distance  s  between  conductor  and  image  conductor  in 
the  case  of  ground  return  is  very  much  larger  —  usually  10  and 
more  times  —  than  the  distance  between  conductor  and  overhead 
return  conductor,  the  inductance  of  a  conductor  with  ground 
return  is  much  larger,  and  the  capacity  smaller,  than  that  of  the 
same  conductor  with  overhead  return.  In  the  former  case,  how- 
ever, this  inductance  and  capacity  are  those  of  the  entire  circuit, 
since  the  ground  return,  as  conducting  plane,  has  no  inductance 
and  capacity;  while  in  the  case  of  overhead  return,  the  inductance 
of  the  entire  circuit  of  conductor  and  return  conductor  is  twice, 
the  capacity  half,  that  of  a  single  conductor,  and  therefore  the 
total  inductance  of  a  circuit  of  two  overhead  conductors  is  greater, 


ROUND   PARALLEL   CONDUCTORS.  131 

the  capacity  less,  than  that  of  a  single  conductor  with  ground 
return. 

The  conception  of  the  image  conductor  is  based  on  that  of  the 
ground  as  a  conducting  plane  of  perfect  conductivity,  and  assumes 
that  the  return  is  by  a  current  sheet  at  the  ground  surface.  As 
regards  the  capacity,  this  is  probably  almost  always  the  case,  as 
even  dry  sandy  soil  or  rock  has  sufficient  conductivity  to  carry, 
distributed  over  its  wide  surface,  a  current  equal  to  the  capacity 
current  of  the  overhead  conductor.  With  the  magnetic  field,  and 
thus  with  the  inductance,  this  is  not  always  the  case,  but  the  con- 
ductivity of  the  soil  may  be  much  below  that  required  to  conduct 
the  return  current  as  a  surface  current  sheet.  If  the  return  cur- 
rent penetrates  to  a  considerable  depth  into  the  ground,  it  may 
be  represented  approximately  as  a  current  sheet  at  some  distance 
below  the  ground,  and  the  "image  conductor  "  then  is  not  the 
image  of  the  overhead  conductor  below  ground,  but  much  lower; 
that  is,  the  distance  s  in  the  equation  of  the  inductance  is  more, 
and  often  much  more,  than  twice  the  distance  of  the  overhead 
conductor  above  ground.  However,  even  if  the  ground  is  of 
relatively  low  conductivity,  and  the  return  current  thus  has  to 
penetrate  to  a  considerable  distance  into  the  ground,  the  induc- 
tance of  the  overhead  conductor  usually  is  not  very  much  increased, 
as  it  varies  only  little  with  the  distance  s.  For  instance,  if  the 
overhead  conductor  is  J  inch  diameter  and  25  feet  above  ground, 
then,  assuming  perfect  conductivity  of  the  ground  surface,  the 
inductance  would  be 


and 


r  =  i";  s  =  2  X  25'  =  600",  hence  -  =  2400, 


L  =  2  ]  log  -  +       10~9  =  16.066  X  10~9  h. 

T        Z  \ 


If,  however,  the  ground  were  of  such  high  resistance  that  the  cur- 
rent would  have  to  penetrate  to  a  depth  of  over  a  hundred  feet, 
and  the  mean  depth  of  the  ground  current  were  at  50  feet,  this 

would  give  s  =  2  X  75'  =  1800",  hence  -  =  7200,  and 

L  =  18.264  X  10-9  h, 
or  only  13.7  per  cent  higher.     In  this  case,  however,  the  ground  sec- 


132      ELECTRIC  DISCHARGES,   WAVES  AND  IMPULSES. 

tion  available  for  the  return  current,  assuming  its  effective  width 
as  800  feet,  would  be  80,000  square  feet,  or  60  million  times 
greater  than  the  section  of  the  overhead  conductor. 

Thus  only  with  very  high  resistance  soil,  as  very  dry  sandy  soil, 
or  rock,  can  a  considerable  increase  of  the  inductance  of  the  over- 
head conductor  be  expected  over  that  calculated  by  the  assump- 
tion of  the  ground  as  perfect  conductor. 

F.   Mutual  induction  between  circuits. 

48.  The  mutual  inductance  between  two  circuits  is  the  ratio 
of  the  current  in  one  circuit  into  the  magnetic  flux  produced  by 
this  current  and  interlinked  with  the  second  circuit.  That  is, 

j        _  $2  _  $1 

Lim  —  ~  --  -r  i 
ll  li 

where  $2  is  the  magnetic  flux  interlinked  with  the  second  circuit, 
which  is  produced  by  current  i\  in  the  first  circuit. 

.  In  the  same  manner  as  the  self-inductance   L, 

the  mutual  inductance  Lm  between  two  circuits  is 

calculated;  while  the  (external)  self-inductance  cor- 

°  B  responds  to   the   magnetic   flux   between   the    dis- 

tances r  and  s,  the  mutual  inductance  of  a  conductor 

k   a    A  upon  a  circuit  ab  corresponds  to  the  magnetic  flux 

0    °     produced  by  the  conductor  A  and  passing  between 

Fig-  65'         the  distances  Aa  and  Ab,  Fig.  65. 

Thus  the  mutual  inductance  between  a  circuit  AB  and  a  circuit 

ab  is  mutual  inductance  of  A  upon  ab, 


Jiutual  inductance  of  B  upon  ab, 


hence  mutual  inductance  between  circuits  AB  and  ab, 
Lm  =  Lm"  —  Lm  , 


where  A  a,  Ab,  Ba,  Bb  are  the  distances  between  the  respective 
conductors,  as  shown  in  Fig.  66. 


ROUND   PARALLEL  CONDUCTORS. 


133 


If  one  or  both  circuits  have  ground  return,  they  are  replaced 
by  the  circuit  of  the  overhead  conductor  and  its  image  conductor 
below  ground,  as  discussed  before. 

If  the  distance  D  between  the  circuits 
AB  and  ab  is  great  compared  to  the  dis- 
tance S  between  the  conductors  of  circuit 
A  B,  and  the  distance  s  between  the  con- 
ductors of  circuit  ab,  and  0  =  angle  which 
the  plane  of  circuit  AB  makes  with  the 
distance  D,  ty  the  corresponding  angle  of 
shown  in  Fig.  66,  it  is 


circuit    a&,    as 
approximately 


Fig.  66. 


Aa  =  D  -f-  £  cos  0  +  -  cos 
Ab  =  D  +  —  cos  0  —  -  cos 

A  A 

Ba  =  D  —  —  cos  0  -{-  ~  cos 

2i  2i 

Bb  =  D  —  -  cos  0  —  ^  cos 


(42) 


hence 


m  =  21og- 


n      , 

D+ 


2  log 


D2  -  I-  cos 0  -  ~ cos 

D2-  (7:COS0  -fxCOS 


=  2 


log 


2  COS  0  —  jz  COS 


-  log   1  - 


x  io~s  /?, 


hence  by  ( 

T            __     rt 

18) 

PC  s 

s          \2      AS              s 

)2 

D2 

134      ELECTRIC  DISCHARGES,    WAVES  AND  IMPULSES. 

thus 

2  **!()-.*.  (43) 


For  0  =  90  degrees  or  ty  =  90  degrees,  Lm  is  a  minimum,,  and 
the  approximation  (43)  vanishes. 

G.   Mutual  capacity  between  circuits. 

49.  The  mutual  capacity  between  two  circuits  is  the  ratio  of 
the  voltage  between  the  conductors  of  one  circuit  into  the  dielec- 
tric flux  produced  by  this  voltage  between  the  conductors  of  the 
other  circuit.  That  is 


where  ^2  is  the  dielectric  flux  produced  between  the  conductors 
of  the  second  circuit  by  the  voltage  e\  between  the  conductors  of 
the  first  circuit. 

If  e  =  voltage  between  conductors  A  and  B,  the  dielectric  flux 
of  conductor  A  is,  by  (36), 

t  =  Ce  =  -          ,  (44) 


where  R  is  the  radius  of  these  conductors  and  S  their  distance 
from  each  other. 

This  dielectric  flux  produces,  by  (32),  between  the  distances  Aa 
and  A  b,  the  potential  difference 

Aa 

g' 


and  the  dielectric  flux  of  conductor  B  produces  the  potential 
difference 

2v2-fr,      Ba.  /A0* 

e    =  -  — l°g^r>  (4w 

K  no 

hence  the  total  potential  difference  between  a  and  b  is 

2v2iK      AbBa. 


substituting  (44)  into  (47), 

e  Ab  Ba 


ROUND  PARALLEL  CONDUCTORS.  135 

and  the  dielectric  flux  produced  by  the  potential  difference  e"  —  ef 
between  the  conductors  a  and  b  is 

.  K€          ,     Ab  Ba 


2  v2  log-  log  ^ 


hence  the  mutual  capacity 

K 


2  v2  log  -  log  — 


or,  by  approximation  (18),  as  in  (43), 

Cm=  «&«*»*«»*  1(y  ,.  (49) 


This  value  applies  only  if  conductors  A  and  B  have  the  same 
voltage  against  ground,  in  opposite  direction,  as  is  the  case  if 
their  neutral  is  grounded. 

If  the  voltages  are  different,  e\  and  e2,  where  e\  +  e2  =  2  e,  as 
for  instance  one  conductor  grounded: 

ei  =  0,  62  =  e,  (50) 

the  dielectric  fluxes  of  the  two  conductors  are  different,  and  that 
of  A  is:  crt/r;  that  of  B  is:  c2^,  where 


=  f2. 

2       e ' 

and 

d  +  c2  =  2, 

the  equations  (45)  to  (49)  assume  the  forms; 

Aa 


K  AO 

2  v2^ ,     Ba 


(52) 
(53) 

//  /  —  (      Y1"     i          i          j  s'<  -,          ./I rt    / 

e"  -  e'  =  -  -^  j  c2  log  BT  -  ci  log^r  [  ,  (54) 

ir  r*\r\  /\  r\    \  %    -   <f 


136      ELECTRIC  DISCHARGES,  WAVES  AND  IMPULSES. 

Cm  = -,   j  C2  log^  -  c,  log^  |  lO-9/ 

«  -i-    *i  •  »  I  Bo  Ab  \ 


2  v2  log  -  log  -5 


(55) 


y  £42): 


Ba  Aa 

~5l  -cilogT7 


COS  0  —  S  COS 


+  SCOS' 


+ 


ir^-)- 


COS  0  —  S  COS 


2D 


and  this  gives : 

fe  —  Ci)  s  cos 


D 


+ 


+  Cfc)  /Ss  COS  0 


(56) 


hence 

c.=- 


x  .  S  COS  ^    ,     /  N 

fe-cO—^+fe  +  d)         2D, 


and  for  61  =  0,  and  thus  c\  =  0,  c2  =  2: 


COS 


+ 


cos  <t> 


9  n 

^  LJ 


io9/, 


(57) 
(58) 


hence  very  much  larger  than  (49).  However,  equation  (58) 
applies  only,  if  the  ground  is  at  a  distance  very  large  compared 
with  Z),  as  it  does  not  consider  the  ground  as  the  static  return  of 
the  conductor  B. 

H.    The  three-phase  circuit. 

50.   The  equations  of  the  inductance  and  the  capacity  of  a 
conductor 

(26) 


109/ 


(37) 


ROUND   PARALLEL  CONDUCTORS.  137 

apply  equally  to  the  two-wire  single-phase  circuit,  the  single  wire 
circuit  with  ground  return,  or  the  three-phase  circuit. 
In  the  expression  of  the  energy  per  conductor: 

Li' 

(59) 


and  of  the  inductance  voltage  e'  and  capacity  current  i',  per 
conductor: 

'  =  (60) 

i  is  the  current  in  the  conductor,  thus  in  a  three-phase  system  the 
Y  or  star  current,  and  e  is  the  voltage  per  conductor,  that  is,  the 
voltage  from  conductor  to  ground,  which  is  one-half  the  voltage 

between  the  conductors  of  a  single-phase  two-wire  circuit,  —T=-  the 

voltage  between  the  conductors  of  a  three-phase  circuit  (that  is, 
it  is  the  Y  or  star  voltage),  and  is  the  voltage  of  the  circuit  in  a 
conductor  to  ground,  s  is  the  distance  between  the  conductors,  and 
is  twice  the  distance  from  conductor  to  ground  in  a  single  con- 
ductor with  ground  return.* 

If  the  conductors  of  a  three-phase  system  are  arranged  in  a 
triangle,  s  is  the  same  for  all  three  conductors;  otherwise  the 
different  conductors  have  different  values  of  s,  and  A    B     c 
the  same  conductor  may  have  two  different  values  of  °     °     o 
s,  for  its  two  return  conductors  or  phases. 

For   instance,    in  the  common  arrangement  of  the  o^ 

three-phase  conductors   above   each    other,   or  beside 
each  other,  as  shown  in  Fig.  67,  if  s  is  the  distance 
between  middle  conductor  and  outside  conductors,  the          OQ 
distance  between  the  two  outside  conductors  is  2  s.         Fig.  67. 

The  inductance  of  the  middle  conductor  then  is: 

(61) 

The  inductance  of  each  of  the  outside  conductors  is,  with  respect 
to  the  middle  conductor: 

*  See  discussion  in  paragraph  47. 


138      ELECTRIC  DISCHARGES,   WAVES  AND  IMPULSES. 

(62) 
With  respect  to  the  other  outside  conductor: 

L  =  2Jlogy  +  ^jlO-U.  (63) 

The  inductance  (62)  applies  to  the  component  of  current, 
which  returns  over  the  middle  conductor,  the  inductance  (63), 
which  is  larger,  to  the  component  of  current  which  returns  over 
the  other  outside  conductor.  These  two  currents  are  60  degrees 
displaced  in  phase  from  each  other.  The  inductance  voltages, 
which  are  90  degrees  ahead  of  the  current,  thus  also  are  60  degrees 
displaced  from  each  other.  As  they  are  unequal,  their  resultant 
is  not  90  degrees  ahead  of  the  resultant  current,  but  more  in  the 
one,  less  in  the  other  outside  conductor.  The  inductance  voltage 
of  the  two  outside  conductors  thus  contains  an  energy  component, 
which  is  positive  in  the  one,  negative  in  the  other  outside  conductor. 
That  is,  a  power  transfer  by  mutual  inductance  occurs  between 
the  outside  conductors  of  the  three-phase  circuit  arranged  as  in 
Fig.  67.  The  investigation  of  this  phenomenon  is  given  by 
C.  M.  Davis  in  the  Electrical  Review  and  Western  Electrician  for 
April  1,  1911. 

If  the  line  conductors  are  transposed  sufficiently  often  to  average 
their  inductances,  the  inductances  of  all  three  conductors,  and  also 
their  capacities,  become  equal,  and  can  be  calculated  by  using  the 
average  of  the  three  distances  s,  s,  2  s  between  the  conductors, 

4  s 

that  is,  -  s,  or  more  accurately,  by  using  the  average  of  the  log  -  > 

o  r 

s  2s 

log  -  and  log  -5-  ,  that  is: 
r  o 


•  3 

In  the  same  manner,  with  any  other  configuration  of  the  line 
conductors,  in  case  of  transposition  the  inductance  and  capacity 

Q 

can  be  calculated  by  using  the  average  value  of  the  log  -  between 

the  three  conductors. 

The  calculation  of  the  mutual  inductance  and  mutual  capacity 
between  the  three-phase  circuit  and  a  two-wire  circuit  is  made 


ROUND  PARALLEL  CONDUCTORS.  139 

in  the  same  manner  as  in  equation  (41),  except  that  three  terms 
appear,  and  the  phases  of  the  three  currents  have  to  be  con- 
sidered. Q^ 

Thus,  if  A,  B,  C  are  the  three  three-phase  con- 
ductors, and  a  and  b  the  conductors  of  the  second 
circuit,  as  shown  in  Fig.  68,  and  if  ii,  iz,  i3  are  °C        OB 
the  three  currents,  with  their  respective  phase 
angles   71,   72,   73,    and  i   the   average   current,  b     a 

denoting: 


o 

Fig.  68. 

1\  12  ^3 


conductor  A  gives: 
conductor  B: 

conductor  C: 

Lm'"  =  2  c3  cos  (0  -  240°  -  73)  log^?>* 
hence, 

Lm  =  2 )  ci  cos  03  -  71)  log  4r  +  C2  cos  08  -  120°  -  72)  log  |?, 


Lm"  =  2  c2  cos  08  -  120°-  72)  log!?, 

no 


4-  c3  cos  (0  -  240°-  73)  log  ^  |  10-9  /i, 


and  in  analogous  manner  the  capacity  Cm  is  derived. 

In  these  expressions,  the  trigonometric  functions  represent  a 
rotation  of  the  inductance  combined  with  a  pulsation. 


INDEX. 


PAGE 

Acceleration  as  mechanical  transient 4 

single-energy  transient 8 

Admittance,  natural  or  surge,  of  circuit 61,  84 

Alternating  current  in  line  as  undamped  oscillation 97 

phenomena  as  transients 9 

reduction  to  permanents 9 

Alternators,  momentary  short-circuit  currents 37 

construction 40 

calculation  44 

Arcing  grounds 97 

Armature  transient  of  alternator  short  circuit 41 

Attenuation  of  transient,  see  Duration. 

Cable  inductance,  calculation 81,  123 

surge 62 

Capacity 18 

calculation 127 

of  circuit,  definition 12 

current 13 

definition 119 

effective,  of  line  transient 75 

equation 129 

and  inductance  calculation  of  three-phase  circuit 136 

CAPACITY    AND     INDUCTANCE     OF    ROUND     PARALLEL 

CONDUCTORS 119 

Capacity,  mutual,  calculation .T 134,  138 

specific 16,  17,  18 

Charge,  electric,  of  conductor 14 

Charging  current 13 

Circuit,  dielectric 14,  17,  18 

of  distributed  capacity  and  inductance,  also  see  Line. 

electric 17,  18 

magnetic 14,  17,  18 

Closed  compound-circuit  transient Ill,  112 

Combination  of  effective  and  reactive  power 100 

transient 100 

of  standing  and  traveling  waves 100 

COMPOUND   CIRCUIT  OSCILLATION 108 

Compound  circuit,  power  flow 90 

velocity  unit  of  length 92 

oscillation  of  closed  circuit Ill,  112 

of  open  circuit 0 114 

141 


142  INDEX. 

PAGE 

Condenser  current 13 

Conductance 18 

effective,  of  line  transient 78 

Conductivity,  electric 18 

Cumulative  oscillation 97 

Current,  electric 18 

in  field  at  alternator  short  circuit 40 

transient  pulsation 43 

permanent  pulsation 45 

transient,  maximum 61 

Danger  from  single-energy  magnetic  transient 27 

Decay  of  single-energy  transient 21 

Deceleration  as  mechanical  transient 4 

Decrease  of  transient  energy 59 

Decrement  of  distance  and  of  time 94 

exponential 88 

Decrease  of  power  flow  in  traveling  wave 92 

Density,  dielectric 16,  17,  18 

electric  current 18 

magnetic 15,  17,  18 

Dielectric  field 11 

as  stored  energy 3 

forces 10 

flux 15,  17,  18 

gradient 18 

transient,  duration 59 

Dielectrics 15,  17,  18 

Disruptive  effects  of  transient  voltage 63 

Dissipation  constant  of  circuit 94 

compound  circuit 109 

double-energy  transient 68 

line 78 

dielectric  energy  in  double  energy-transient 67 

exponent  of  double-energy  transient 68 

of  magnetic  energy  in  double-energy  transient 67 

Distortion  of  quadrature  phase  in  single-phase  alternator  short  circuit . .  47 

Distance  decrement 94 

Distributed  capacity  and  inductance 73 

Double-energy  transient 7 

equation 69 

DOUBLE-ENERGY  TRANSIENTS 59 

Double  frequency  pulsation  of  field  current  at  single-phase  alternator 

short  circuit 45 

Duration  of  double-energy  transient 68 

single-energy  transient 22,  27 

transient 59 

alternator  short-circuit  current 41 


INDEX.  143 

PAGE 

Effective  values,  reducing  A.C.  phenomena  to  permanents 9 

Elastance 18 

Elastivity 18 

Electrifying  force 15,  17 

Electromotive  force 15,  17,  18 

Electrostatic,  see  Dielectric. 

Energy,  dielectric 18 

of  dielectric  field 13 

dielectric  and  magnetic,  of  transient 67 

magnetic 18 

of  magnetic  field 12 

storage  as  cause  of  transients 3 

transfer  in  double-energy  transient 60 

by  traveling  wave 92 

of  traveling  wave  in  compound  circuit 110 

Equations  of  double-energy  transient 69 

line  oscillation 74,  75}  84 

simple  transient 6 

single-energy  magnetic  transient 21,  24 

Excessive  momentary  short  circuit  of  alternator 37 

Exponential  decrement 88 

magnetic  single-energy  transient 21 

transient 7 

numerical  values.  ..'..- 23 


Field  current  at  alternator  short  circuit,  rise 40 

transient  pulsation 43 

permanent  pulsation 45 

FIELD,  ELECTRIC 10 

rotating,  transient 34 

superposition 120 

transient,  of  alternator 38 

construction 40 

calculation 44 

Flux,  dielectric 11,  15 

magnetic 10,  14 

Frequencies  of  line  oscillations 79 

Frequency  of  double-energy  transient,  calculation 66 

oscillation  of  line  transient 78 

Frohlich's  formula  of  magnetic-flux  density 53 

Fundamental  wave  of  oscillation .  .  81 


Gradient,  electric 15,  17,  18 

Grounded  phase 97 

Grounding  surge  of  circuit 62 

Ground  return  of  conductor,  inductance  and  capacity 130 


144  INDEX. 

PAGE 

Half- wave  oscillation 82 

Hunting  of  synchronous  machines  as  double-energy  transient 9 

Hydraulic  transient  of  water  power 4 

Image  conductor  of  grounded  overhead  line 130 

Impedance,  natural  or  surge,  of  circuit 61,  84 

Impulse  propagation  over  line  and  reflection 105 

as  traveling  wave 105 

Increase  of  power  flow  in  traveling  wave 92 

Independence  of  rotating-field  transient  from  phase  at  start 36 

Inductance  of  cable 123 

calculation 123 

and  capacity  calculation  of  three-phase  circuit 136 

INDUCTANCE     AND     CAPACITY     OF    ROUND     PARALLEL 

CONDUCTORS 119 

Inductance  of  circuit,  definition 11 

definition 119 

effective,  of  line  transient 75 

equation 123,  126,  131 

mutual,  calculation 132,  138 

voltage 12,  18 

Intensity,  dielectric 16,  17,  18 

magnetic 15,  17,  18 

IRONCLAD  CIRCUIT,  SINGLE-ENERGY  TRANSIENT 52 

Ironclad  circuit  transient,  oscillogram 57 

Kennelly's  formula  of  magnetic  reluctivity 53 

Length  of  circuit  in  velocity  measure,  calculation 109 

Lightning  surge  of  circuit 62 

as  traveling  wave 89 

Line  as  generator  of  transient  power 112 

LINE  OSCILLATION 72 

general  form 74 

also  see  Transmission  line. 

Magnetic  field 10 

as  stored  energy 3 

flux 14,  17,  18 

forces 10 

single-energy  transient 19,  25 

construction 20,  25 

duration 59 

Magnetics 14,  17,  18 

Magnetizing  force 14,  17,  18 

Magnetomotive  force 14,  17,  18 

Massed  capacity  and  inductance 73 

Maximum  transient  current 61 

voltage 61 


INDEX.  145 

PAGE 

Maximum  value  of  rotating-field  transient 36 

Measurement  of  very  high  frequency  traveling  wave 104 

Mechanical  energy  transient 4 

Momentary  short-circuit  current  of  alternators 37 

construction 40 

calculation 44 

Motor  field,  magnetic  transient 24 

Mutual  capacity,  calculation 134,  138 

of  lines,  calculation 81 

inductance  and  capacity  with  three-phase  circuit 138 

calculation 132,  138 

Natural  admittance  and  impedance  of  circuit 61,  84 

Nonperiodic  transient 9 

Nonproportional  electric  transient 52 

surge  of  transformer 64 

Open-circuit  compound  oscillation 113 

Oscillating  currents. 62 

voltages 62 

Oscillation  frequency  of  line  transient • 78 

Oscillation  of  open  compound  circuit 113 

stationary,  see  Stationary  oscillations  and  standing  waves. 

Oscillations,  cumulative 97 

of  closed  compound  circuit Ill,  112 

OSCILLATIONS,  LINE 72 

OSCILLATIONS  OF  THE  COMPOUND   CIRCUIT 108 

Oscillatory  transient  of  rotating  field 36 

Oscillograms  of  arcing  ground  on  transmission  line 98 

cumulative  transformer  oscillation 99 

decay  of  compound  circuit 91 

formation  of  stationary  oscillation  by  reflection  of  traveling 

wave 101 

high-frequency   waves   preceding   low-frequency  oscillation  of 

compound  circuit 102,  103 

impulses  in  line  and  their  reflection 106 

single-phase  short  circuit  of  alternators 50 

single-phase  short  circuit  of  quarter-phase  alternator 48 

three-phase  short  circuit  of  alternators 49 

starting  current  of  transformer 57 

starting  oscillation  of  transmission  line 76,  77 

varying  frequency  transient  of  transformer 64 

Pendulum  as  double-energy  transient 8 

Periodic  component  of  double-energy  transient,  equation 66 

energy  transfer  in  transient 60 

and  transient  component  of  transient 72 

transients,  reduction  to  permanents 9 


146  INDEX. 

PAGE 

Period  and  wave  length  in  velocity  units 92 

Permanent  phenomena,  nature 1 

Permeability 15,  17,  18 

Permeance 18 

Permittance 18 

Permittivity 16,  17,  18 

Phase  angle,  change  at  transition  point 117 

of  oscillation,  progressive  change  in  line 75 

Phenomena,  transient,  see  Transients. 

Polyphase  alternator  short  circuit 44 

oscillograms 48,  49 

Power  diagram  of  open  compound-circuit  transient 114 

closed  compound-circuit  transient Ill,  113 

dissipation  constant 88 

of  section  of  compound  circuit 108 

double-energy  transient 66 

electric 18 

flow  in  compound  circuit 90 

of  'line  transient 88,  89 

of  line  oscillation 79 

transfer  constant  of  circuit 94 

sectiDn  of  compound  circuit 108 

in  compound-circuit  oscillation 90 

of  traveling  wave 95 

Progressive  change  of  phase  of  line  oscillation 75 

Propagation  of  transient  in  line 74 

velocity  of  electric  field 74 

field 129 

Proportionality  in  simple  transient 4 

Pulsation,  permanent,  of  field  current  in  single-phase  alternator  short 

circuit 45 

of  transient  energy 61 

transient,  of  magnetomotive  force  and  field  current  at  poly- 
phase alternator  short  circuit 41 

Quadrature  relation  of  stationary  wave 88 

Quantity  of  electricity 14 

Quarter-wave  oscillation  of  line 81,  82 

Reactance  of  alternator,  synchronous  and  self-inductive 37 

Reaction,  armature,  of  alternator 37 

Reactive  power  wave 88 

Reflected  wave  at  transition  point 118 

Reflection  at  transition  point 118 

Relation  between  capacity  and  inductance  of  line 81 

standing  and  traveling  waves 101 

Reluctance 18 

Reluctivity 18 


INDEX.  147 

PAGE 

Resistance 18 

effective,  of  line  transient 78 

Resistivity 18 

Resolution  into  transient  and  permanent 30 

Rise  of  field  current  at  alternator  short  circuit 40 

Rotating  field,  transient 34 

Self-induction,  e.m.f.  of 12 

Separation  of  transient  and  permanent 27,  30 

Ships;  deceleration  as  transient 7 

Short-circuit  current  of  alternator,  momentary 37 

construction 40 

calculation 44 

surge  of  circuit 62 

Simple  transient 4 

equation 6 

Single-phase  alternator  short  circuit 45 

oscillogram 50 

short  circuit  of  alternators 45 

oscillograms 48,  50 

Single-energy  transient . 7 

SINGLE-ENERGY  TRANSIENT  OF  IRON-CLAD  CIRCUITS.  52 

SINGLE-ENERGY  TRANSIENTS,   CONTINUOUS   CURRENT.  19 

SINGLE-ENERGY  TRANSIENTS  IN  A.C.  CIRCUITS 30 

Specific  capacity 16,  17,  18 

Standing  waves 97 

originating  from  traveling  waves 101 

see  Stationary  oscillation  and  Oscillation,  stationary. 

Start  of  standing  wave  by  traveling  wave 101 

Starting  current  of  transformer,  oscillogram ^ 57 

oscillation  of  line 86 

transmission  line,  oscillogram 76,  77 

of  rotating  field 34 

transient  of  A.C.  circuit 32 

magnetic  circuit 28 

three-phase  circuit 32 

Static  induction  of  line,  calculation 81 

Stationary  oscillation  of  open  line 86 

see  Line  oscillation  and  Standing  wave. 

Steepness  of  ironclad  transient 58 

wave  front  and  power-transfer  constants  of  impulse 105 

Step-by-step  method  of  calculating  transient  of  ironclad  circuit 53 

Storage  of  energy  as  cause  of  transients 3 

Superposition  of  fields 120 

Surge  admittance  and  impedance  of  circuit 61,  84 

Symbolic  method  reducing  A.C.  phenomena  to  permanents 9 

Symmetrical  pulsation  of  field  current  at  single-phase  alternator  short 

circuit .  .                                                      45 


148  INDEX. 

PAGE 

Terminal  conditions  of  line  oscillation 84 

Three-phase  alternator  short  circuit 44 

oscillograms 49 

Three-phase  circuit,  inductance  and  capacity  calculations 136 

mutual  inductance  and  capacity 138 

current  transient 32 

.-.    magnetic-field  transient 34 

Time  constant 22 

decrement 94 

of  compound  circuit 108 

Transfer  of  energy  in  double-energy  transient 60 

by  traveling  wave 92 

in  compound  circuit  oscillation 90 

Transformation  ratio  at  transition  point  of  compound  circuit 117 

Transformer  as  generator  of  transient  energy Ill,  113 

Transformer  surge 62 

Transient  current  in  A.C.  circuit 31 

double-energy 7 

power  transfer  in  compound-circuit 112 

single-energy 7 

of  rotating  field 34 

separation  from  permanent 27,  30 

short-circuit  current  of  alternator 37 

construction 40 

calculation 44 

and  periodic  components  of  oscillation   72 

Transients,  double-energy 59 

caused  by  energy  storage 3 

fundamental  condition  of  appearance 4 

general  with  all  forms  of  energy 4 

as  intermediate  between  permanents 2 

nature 1 

single-energy  A.C.  circuit 30 

continuous  current 19 

Transition  point,  change  of  phase  angle 117 

reflection 118 

voltage  and  current  transformation 117 

Transmission-line  surge 63 

transient 73 

also  see  Line. 

Transmitted  wave  at  transition  point 118 

Transposition  of  line  conductors 138 

TRAVELING  WAVES 88 

Traveling  wave,  equation 95 

as  impulses 105 

preceding  stationary  oscillation 101 

of  very  high  frequency 104 


INDEX.  149 

PAGE 

Turboalternators,  momentary  short-circuit  current 37 

construction 40 

calculations 44 

Undamped  oscillations 97 

Unidirectional  energy  transfer  in  transient 60 

Uniform  power  flow  in  traveling  wave 92 

Unsymmetrical  pulsation  of  field  current  at  single-phase  alternator  short 

circuit 46 

Varying  frequency  oscillation  of  transformer 64 

Velocity  constant  of  line 78,  83 

relation  between  line  capacity  and  inductance 81 

measure,  calculation  of  circuit  length 109 

of  propagation  of  electric  field 74,  129 

transient  of  ship 7 

unit  of  length  of  line 83 

of  length  in  compound  circuit 92 

Very  high  frequency  traveling  wave 104 

Voltage 18 

gradient .  15,  17 

relation  of  section  of  compound  oscillating  circuit 17 

rise  of  quadrature  phase  in  single-phase  alternator  short  circuit      47 

at  transition  point  of  compound  oscillation 117 

of  single-energy  magnetic  transient 27 

transient,  maximum 61 

Wave  front  of  impulse 105 

length  of  line  oscillation 79,  80 

and  period  in  velocity  unit 92 

standing,  see  Standing  wave. 

WAVES,  TRAVELING 88 


TOO 


i 


33417 


749213 


753 


Engineering 
Library 

UNIVERSITY  OF  CALIFORNIA  LIBRARY 


